Can you choose the order of effects when playing a Tag Team Supporter, or do you have to do them in the order stated on the card?
You must do them in the order stated on the card. However, you don't have to meet the criteria for the first effect in order to do the second effect; for example, if the first effect says to search your deck but you don't have any cards in your deck you can still do the second effect as long as you meet its criteria.
Source: Cosmic Eclipse FAQ (2019-10-18), TPCi Rules Team (2019-10-31)