A cheapo used american cameo or similar style press would be ideal, they can be found for under a thou easy. You don't have to use "solvent" inks to get it done, there are waterbased inks from TW that would work great. The thing about this though is that you would probably want to print sheets and you would need the tags to be die cut afterwards. If you have precut tags, that could also work, but I would recommend one of the smaller presses, like a 13x18. I would recommend against singles though since the lack of extra sheet to keep it held down on the table could lead to it sticking to the underside, ink webbing, or just not so great printing.
You can also dry them like a shirt with most inks at around 200F iirc. I just use the traditional drying racks (which can also be found for 250-500 or free if you get lucky) when I print flatstock.
Ryonet claims that you can use their waterbased ink on paper too, I think. Paper can be pretty easy compared to shirts IMO.
I have a little cameo that I roll up to the end of the dryer for single color stuff sometimes, so there's that too.
