I need ideas for a chemical or disease which kills the victim upon holding it in the palm for about 4 or 5 seconds. The death isn't necessarily immediate, as it might take days to kill the victim in the case of an incurable disease.

If the substance can't kill the victim, at least it should cause permanent damage to the heart, nervous system, or brain. An overall skin disfigurement or damage to any other vital organs will be okay!

Something mixed in DMSO?

The specific chemical or disease should be somewhat easy to acquire with some effort for a common man because it must sound obvious for the readers.

Don't suggest nerve agents or Anthrax please. Because they are inaccessible for a common man.

Honeybee.

https://cronkitenews.azpbs.org/2018/02/21/advocacy-groups-pressure-state-to-ban-bee-killing-pesticides/

I like this because if your victim is allergic, he can very definitely die from a bee sting. It takes seconds to get stung. This method is not applicable to the general population, so if user63214 is some disgruntled would-be murderer I am not an accessory. But for a story it could work just fine.

The canonical example is

Dimethylmercury

Dimethylmercury is an organomercury compound. A highly volatile, reactive, flammable, and colorless liquid, dimethylmercury is one of the strongest known neurotoxins, with a quantity of less than 0.1 mL capable of inducing severe mercury poisoning, and is easily absorbed through the skin. Dimethylmercury is capable of permeating many materials, including plastic and rubber compounds. It has a slightly sweet odor, although inhaling enough of the chemical to notice this would be hazardous. (Wikipedia, s.v. dimethylmercury)

Dimethylmercury is extremely toxic and dangerous to handle. Absorption of doses as low as 0.1 mL can result in severe mercury poisoning. The risks are enhanced because of the high vapor pressure of the liquid.

Permeation tests showed that several types of disposable latex or polyvinyl chloride gloves (typically, about 0.1 mm thick), commonly used in most laboratories and clinical settings, had high and maximal rates of permeation by dimethylmercury within 15 seconds. The American Occupational Safety and Health Administration advises handling dimethylmercury with highly resistant laminated gloves with an additional pair of abrasion-resistant gloves worn over the laminate pair, and also recommends using a face shield and working in a fume hood.

Dimethylmercury is metabolized after several days to methylmercury. Methylmercury crosses the blood–brain barrier easily, probably owing to formation of a complex with cysteine. It is eliminated from the organism slowly, and therefore has a tendency to bioaccumulate. The symptoms of poisoning may be delayed by months, resulting in cases in which a diagnosis is ultimately discovered, but only at the point in which it is too late for an effective treatment regimen to be successful.

The toxicity of dimethylmercury was highlighted with the death of Karen Wetterhahn, a professor of chemistry at Dartmouth College, in 1997. Professor Wetterhahn specialized in heavy metal poisoning. After she spilled a few drops of this compound on her latex glove, the barrier was compromised, and within minutes it was absorbed into her skin. It circulated through her body and accumulated in her brain, resulting in her death ten months later. This accident is a common toxicology case-study and directly resulted in improved safety procedures for chemical-protection clothing and fume use, which are now called for when any exposure to such severely toxic and/or highly penetrative substances is possible (e.g., in chemical munitions stockpiles and decontamination facilities).

The beauty of it is that dimethylmercury is easily synthesized:

The compound was one of the earliest organometallics reported, reflecting its considerable stability. It is formed by treating sodium amalgam with methyl halides:

$mathrmHg + 2 mathrmNa + 2 mathrm CH_3mathrmI rightarrow mathrm Hg(mathrmCH_3)_2 + 2 mathrm NaI$

Methyl iodide ($mathrm CH_3mathrmI$) itself is also easy to make.

All in all, all you need in order to make one the deadliest known substances is access to a high-school chemistry laboratory.

While i do not know of any easily obtained chemicals or substances that can kill as you just through contact of your palm (probably a good thing for the general population) i can offer you some ideas.

Delivery

Firstly though, i am going to present you with a possible delivery system. As many chemicals need to be in your bloodstream to take effect, we would need some way of transferring them there. Thusly, i present you with The Assassin’s Glove (name pending).

This would be a glove who’s palm is laced with hypodermic needles, similar to those you may find in nettles. As you shook someone’s hand, these needles would be pressed against their palm and inject the substance into their body. With this glove, you could inject practically anything into the target. Here are some suggestions:

Substances

Snake venom is relatively easy to obtain, assuming you can find and extract the venom from one. However, as it is a venom, it can’t kill you unless it enters your blood stream. You can safely drink snake venom if you do not have any cuts from your mouth to your stomach. However, if you had a cut or it was injected into you, then you would feel its effects.

Cyanide is a classic killer, you can extract it from apple seeds if your protagonist truly has no access to any kind of harmful substance.

Hydrochloric acid, extracted from bleach, may also do the job. Bleach does have a strong smell to it though which may give your assassin away.

Alternatively

If you’re just looking for a way to stealthily kill someone, polonium has famously been used. It is a radioactive substance which emits alpha radiation. Alpha radiation can not penetrate human skin but, of ingested, it is fatal. The reason it kills you is the same reason you are normally safe from it, it can’t penetrate your skin, meaning its stuck inside you and damages your DNA. It would likely take weeks or months to kill someone this way but it has been used in the past. You might also be able to inject it to get a similar effect.

Elemental Mercury

Okay, this doesn't really meet your requirements. Not much is absorbed through the skin. But mercury in your hand is fun to play with and your victim might keep it on longer than 4-5 seconds. The danger is from the vapors. If your plot allows, pouring large amounts of mercury around the victim's bed or car, or another relatively small space where the victim spends a lot of time, might do the trick.

Elemental mercury is what's in thermometers (the kind without batteries), so it would be relatively simple (though expensive and time consuming) to gather enough for the perpetrator to use (assuming a lot of safety equipment, the kind easily found online). The more dangerous form of mercury is methyl mercury, but that's not exactly sold in handy dandy glass containers. Mercury also has medical applications but purchasing it in larger quantities might prove more difficult.

Instead of buying old fashioned thermometers, the perp might want to buy new digital ones and offer them free to doctor's clinics and households and schools in exchange for mercury ones (as a way to reduce potential toxic exposures...swap programs like this probably already exist legitimately). You may also have some luck with fluorescent light bulbs.

Elemental mercury, also called “quicksilver,” is a heavy, silvery,
form of the metal mercury that is liquid at room temperature. It can
slowly change from a liquid into a gas that is invisible to the naked
eye. The gas or “vapors” that are released will start to fill a room
if mercury is spilled indoors.

Mercury is a very
toxic or poisonous substance that people can be exposed to in several
ways. If it is swallowed, like from a broken thermometer, it mostly
passes through your body and very little is absorbed. If you touch it,
a small amount may pass through your skin, but not usually enough to
harm you. Mercury is most harmful when you breathe in the vapors that
are released when a container is open or a spill occurs. (ref)

Will elemental mercury fumes kill a victim? Maybe. But organ damage (which you allow as an alternative) is more likely.

How much mercury spilled in a room will make air in the room unsafe?
Any amount of mercury spilled indoors can be hazardous. The more
mercury is spilled, the more its vapor will build up in air and the
more hazardous it will be. Even a small spill, such as from a broken
thermometer, can produce hazardous amounts of vapor if a room is small
enough, warm enough and people spend a good deal of time there, as in
a small bedroom. (ref)

Spilling the mercury on to a hot surface (like a stove) will vaporize it quickly. Spilling it elsewhere will allow the vapors to accumulate more slowly. The liquid form is easily tracked through a house and hard to get rid of.

Inhalation of elemental mercury vapors is the main source of toxicity,
as mercury is well-absorbed through the lungs. Problems from
inhalation result either from a large one-time high exposure or a
long-term exposure. Long-term exposure of inhaled vapors is generally
more dangerous, with the nervous system being the primary target of
mercury toxicity. Symptoms may occur within weeks but usually develop
insidiously over a period of years. Neurologic symptoms include
tremors, headaches, short-term memory loss, incoordination, weakness,
loss of appetite, altered sense of taste and smell, numbness and
tingling in the hands and feet, insomnia, and excessive sweating.
Psychiatric effects are also seen after long-term exposure. The
kidneys can also be effected. Intense exposure to high concentrations
of mercury vapor can lead to severe respiratory damage. (ref)

So there you have it. Organ damage from even short-term exposures. But not from being held in the hand. Consider that a frame challenge, that the substance you are imagining probably does not exist. But you can combine mercury in the hand with mercury spilling all over the place and creating fumes. Maybe try when the victim is locked into a sauna for an hour. There are many possibilities.