I would research screenprinting shops. You should be able to find some that work on a small scale and use the right inks
You might have some luck looking into sublimation. The feasibility depends on the kind of plastic of the bag, though. I think it should be some kind of polyester.
Other options would need more of an DIY-approach:
- Stencils. You could cut some stencils out of a number of materials and use spraypaint to get your logo on the bag. You might need a primer or test some spraypaint types.
- vinyl cutting. Not sure how it is really called, but it consists of a plotter that cuts adhesive (colored) vinyl sheets into the shapes of your logo. You can stick these to the bags OR use the cutouts as stencils. That way you get the sharpest contours.
These vinyls also react to heat. So it might be that you can ensure a better adhesion to the plastic of the bag by ironing over them.
- Print. Buy some paperstock with autoadhesive backing and print it with your printer. You can then cut the shapes by hand and stick them to the bags. There are laser printers with very good inks that do not run (very important when its wet outside)
Another option might be to find a shop that prints relatively small volumes and order a sticker printed on transparent PVC. If you can get lower than 500 or even 250 you can stick them to your bags and use the rest as promotional handouts/marketing tools/gifts etc..