Yes, you can do that. It says you can: they explicitly mention the 3rd level teamwork feat as an option for getting extra uses.
Do you have to sacrifice a teamwork feat? Is doing so required just to make it once per day?
Nope! By default, just for having the ability, you get one use/day. Note that Determination says you can use it once per day with no catches, no tricks, no trade-off: you just get it, like any other ordinary once per day ability.
At 3rd level, the preacher is a person of few words on the battlefield, but those words hold great power and authority. Once per day, the inquisitor can use this ability to create one of the following effects. Each is a free action to use.
Sacrificing your teamwork feats is something you can optionally do for extra uses. At third level, you receive Determination (which you can use once per day), and a teamwork feat. If you want, you can sacrifice your teamwork feat to be able to use Determination twice per day - or you can just keep the feat.