When designing a t-shirt for my teams, I know the team number needs to be on the shirt, but are there any other requirement for team competition shirts?
Actually, there is no requirements to have competition shirts at all.
It is more a matter of teams exhibiting pride for their organizations. I would look at this topic:
You don’t need to have shirts for your team at a competition, however, if you do, it is more organized, and just better in general. I would include your team name and the student’s name, possibly along with a team logo (only if you have one).
I would take a look at the “share your team shirts” thread to see what other people do.