Is a unit allowed to buy items (unit T-shirts) from a company owned by someone in that unit?

I have to agree with the others that it would probably not be the best business move you would make.

Most of the time when trying to utilize the Government Purchase Card (Government Spending) you will need to get a minimum of three quotes from other vendors prior to submitting the request.  Doesn't mean that your preferred company will have less of an opportunity, but it will protect your organization in allowing other competitors an opportunity.  It will also, show proof that the preferred company is not too expensive for the services rendered  in comparison to the other competition.  It could even show that other companies have a higher cost, making your preferred company save the government money.  (There is a certain dollar threshold per month, so make sure the unit budget officer and the GPC card holder are in the know).

FYI--If the shirts are personalized with Names/Ranks/etc. then they cannot be purchased with government funding.  However if they are generic in nature and available to all members then yes, it could be feasible.  The sticky part is that you cannot sell them in bulk to unit members for profit.  The other option is to check with your unit MWR Funding to see their regulatory guidance.  

Best case, register the unit on post and seek approval for fund raising activities such as car washes, bake sales, local sponsors, etc. to raise the money for the shirts.  As with anything else, you need to have proof in writing before pursuing actions.  (JAG, Command, Installation Agencies).