Vectorize Illustration For Screenprint

I have an illustration that must be converted into a clean, scalable vector file and prepared so it can go straight to my screen-printer without additional tweaks. The artwork will end up on garments, so I need smooth paths, sharp edges, and correct sizing at 300 dpi equivalent... (Budget: $10 - $30 USD, Jobs: Adobe Creative Cloud, Adobe Illustrator, Graphic Design, Illustration, Illustrator, Photoshop, Print Design, Vector Design)