Art does not need to be vector.
I assume your art is physically drawn, like on paper, not in a computer at all, correct?
If it is painterly stuff, it cannot, and should not, be turned into a vector. It should be scanned and turned into a raster image (png, bmp, tiff, etc). Art of this type would probably be printed as simulated process with halftones.
If your art is more along the lines of "lines," then one might scan it and convert the scan to vectors ... but probably best to kept it as a raster file, as above, in order to retain the hand-drawn appearance and save a whole lot of pointless busy work. Art of this sort could probably be printed as spot colors, no halftones required.
Posting a couple of photos will probably get you the best advice. Barring that, say more about the art: medium, style, subject, size, number of colors ... that sort of thing.
Urban America is fairly well infested with copy/fax/shipping stores of various ilks where you can get art, and things, scanned and stuck on a disk. Even items too large for the platen can be scanned in sections and those files aligned in PhotoShop to create the full image. My mom was a painter and did this with quite a bit of her later work. Point being, once you have a decent scan of your art, a competent screen printing shop can turn it into what they need--of course, you'll have to pay them for that work. But if you don't have the tools and/or skills to do that yourself, then you have no choice but to pay someone to prepare the art for you.