If my opponent plays Vanguard Captain to a battlefield they control and I use Gust on it, what happens to the recruit tokens? Does the timing of when I use Gust matter?

The timing of when you use Gust is critical:

If you Gust after the Play Effect resolves: The two Recruit tokens remain at the battlefield. They were already created as separate game objects, and removing Vanguard Captain doesn't affect them. The tokens only cease to exist if they themselves are moved to a non-board zone.

If you Gust in reaction before the Play Effect resolves: The tokens are never created at all. Here's why: When Vanguard Captain is played, it enters the board first, then its Play Effect goes on the chain as a Pending Item. You have a reaction window before this Play Effect resolves. If you Gust Vanguard Captain during this window, it returns to hand before the tokens are created. Since the ability uses location-based targeting ('here'), the source must remain at that location through resolution. When Vanguard Captain changes zones before resolution, the target becomes illegal and the ability doesn't apply, so no tokens are created.