I know both Chewie and the Determination card say to do the ability immediately. So do this allow me to pick which triggers first?

Say Chewie takes a crit would I be able to use Determination to see if it is a pilot trait and discard it, and if not a pilot trait then use Chewie's ability to flip it face down.

Correct.

If the damage card is a critical, it's dealt faceup. That bit doesn't change just because of Chewie. You now have the chance to see what it is. Chewie's ability merely allows you to flip it face down without resolving it, but because Determination and Chewie's ability share the same trigger, you get to choose which one to go with first.

Pilot trait = yes - Determination and discard.

Pilot trait = no - Chewie flips it facedown.