The antimagic field spell says that summoned creatures will temporally vanish until the creature is out of the antimagic field, but would an elemental native to the current plane vanish?

As an example, the UA Mystic can get psionic disciplines that create elementals.

No, they will stay

As you've said, antimagic field (PHB, 213) only affects creatures summoned or created by magic:

Creatures and Objects. A creature or object summoned or created by magic temporarily winks out of existence in the sphere. Such a creature instantly reappears once the space the creature occupied is no longer within the sphere.

Your run-of-the-mill elemental creature type is just a creature in the same way a humanoid, fiend, fey, construct are all just creatures. They would remain while inside an antimagic field.

They will remain, but have no defined physical form

Elementals aren't inherently the product of magic: they exist before a spell is cast, and presumably will exist after its effects fade. However, the embodied form of an element we think of is different from the natural form of elementals. Note the following statement on elementals on page 123 of the Monster Manual (bold added)

On its home plane, an elemental is a bodiless life force. Its dim consciousness manifests as a physical shape only when focused by the power of magic.

Although the existence of elementals is not inherently magical, the familiar form they take on that turns them into creatures whose stat blocks are listed in various books are exclusively the product of some magic or another.

On its own native plane, an elemental isn't a creature summoned by magic. However, the rules on antimagic field state (bold added):

A creature or object summoned or created by magic temporarily winks out of existence in the sphere. Such a creature instantly reappears once the space the creature occupied is no longer within the sphere.

The spirit of the elemental is not created by magic, but the physical form that makes it a defined creature (e.g. an Earth Elemental, an Invisible Stalker, etc.) is created by magic. Thus, that familiar creature will cease to exist in its current form while it is within the antimagic field. Once the field has moved away or ended, the elemental's disembodied spirit will once again form into a physical creature.