I'm sure there indeed is a black market, probably latinum based, for illegal stuff. Kirk's antiques might be gifts though.It really doesn't. How did Kirk acquire all those antiques that decorated his apartment? How does McCoy pay for Romulan Ale?
They may have deluded themselves, hell there may not even be an official Federation currency, but if there isn't, there is one Hell of a Black Market running behind-the-scenes that the Federation looks the other way on.