When a stunned Unit at a battlefield gets stunned again, can vex Vex, Mocking move to that stunned unit's battlefield?

No — Vex, Mocking won't move in that case, because the second stun never happens.

Stunned is a binary state: a Stunned unit can't be Stunned again (Rule 424.1.a.1). The effect can still choose/save-target that unit, but it doesn't become stunned, so no stun event occurs. Vex, Mocking's "When you [Stun] an enemy unit at a battlefield" trigger only fires on an actual stun, so it stays where it is (the Eclipse Herald example under 424.1.a.1 spells this out, and FAQ #4586 says the same for stun-triggered effects).

To move Vex with that ability, the unit has to be un-stunned at the moment the stun effect resolves — so stun a different, not-yet-stunned enemy unit there instead.