You mean shirts like these? http://blankdogtees.com - maybe not quite like these because these are for dogs :)

No, there's no problem with your idea - that's usually how it's done :) Companies have designs and then add their designs to the plain shirts, socks, hats, etc.

You can actually find companies that will even print your name on the tag so it looks like the whole thing came from you :)

Good luck! ;)

~~~~~~~~~~~~~~~