In our area, we have lots of embroidery companies (often "mom and pop" operations) that charge by the piece. My kids are in sports, so we often get monograms on one sports bag or one parka. The only thing is - it's not cheap. We did 2 swim bags last Christmas, and I think it was $11 each. You should be able to find them in the yellow pages under "Embroidery".
With screen printing, the problem is the setup effort. You may just call around and see if you can find someone, but it sounds like you already tried that.
Here are a couple of ideas...
Are you having anything else done, like T-shirts where you might use the same image during production of the higher volume item? (Canvas bags are probably easier than back packs.)
Is the logo something you can do yourself on the computer? I haven't tried this myself, but they make special iron-on transfer paper to be used with a color printer. You print the sheet from your computer and then iron it on.
Do you have a member who is artistic/crafty that might be able to replicate what you want with fabric paint?
I'd probably try the iron on or craft idea using canvas bags myself.
Does anyone know anywhere that you can get just a small quanity of items screen printed or embroidered with your school logo on them?
We are wanting to give away one package per grade level at our Open House that consists of either a backpack or canvas bag with our school logo on it filled with a school tshirt, pencils, etc... all with our logo on them, but I am having a problem finding someone who will do that small of a job (we only need 5 or 6).