Can I reveal a hidden Tideturner in response to a Vex trigger on my unit to move my unit to a battlefield before it gets stunned?

Yes — you can reveal Tideturner from Hidden as a Reaction while Vex's trigger is on the Chain (Hidden grants Reaction, Rule 811.6), and its swap trigger will resolve before Vex's trigger. But it will not save your unit: Vex's trigger targets the unit itself, not a location, so your unit is still a legal target at its new battlefield (Base and Battlefields are all on the Board) and will still be stunned and unable to move this turn. Only leaving the board entirely (hand/trash) would make Vex's trigger mistarget.

One extra wrinkle: playing Tideturner while Vex is at a battlefield triggers Vex a second time targeting Tideturner. Depending on who is the Turn Player, that trigger resolves before or after the swap — either way Tideturner ends up stunned too (and if it's stunned first, per FAQ #10318 your other unit still completes its half of the swap). Rule 423.1.