I'm afraid Kevin's answer is incorrect. It's "All memberS of the team DO their best" but "Every member of the team DOES their best". If "member is in the singular, so is the verb. While the most grammatically correct form is "does his or her best, "does their best" is also accepted to include both genders.

Every member.... everyone... DOES. You treat the expression as singular. "Every member of a team does their best." is correct. (Viviane has explained the use of "their".)

I'm afraid Kevin's answer is incorrect. It's "All memberS of the team DO their best" but "Every member of the team DOES their best". If "member is in the singular, so is the verb. While the most grammatically correct form is "does his or her best", "does their best" is also accepted to include both genders.

The second one ("Every member of a team do their best.") is correct. The first sentence is incorrect because of two things: 1) "Does" should be "do" because "every member" is plural. 2) "Its" should be "his/her" because the team members are people.