Can I use the Inquisitor's (crew) ability to coordinate, when Inqui is on a TIE Reaper with a tactical officer? TO changes coordinate from red to white. Then, Inqui could pick this white action. What do you think? Is it legal?

Yes, it's fine. Actions on upgrade cards are considered to be on your bar.

Sure. GI will change it back to red, of course.

It should be noted that Tactical Officer does not change your red coordinate to white. Rather TO requires that you have a red coordinate (indicated in the restriction box on the lower left) to equip and adds a white coordinate to the ship action bar (indicated by the card's own action bar on the right)

This distinction only matters for fringe cases where you want to use a red action, such as intentionally shutting off ailerons when you're not waiting to use GI. It could be useful to know that you still have both the original red and the upgraded white actions on your bar. Engine Upgrade and Expert Handling work the same way.