Would it be considered tacky/poor taste to give a teacher a $5 or $10 bill with a note saying something like "this is a gift certificate to be used at a store of your choice" instead of something like a mug or bag?

Maybe you could put it is a neat money holder or mug, etc.

I am a teacher and yes we get plenty of mugs etc. Just a nice card with the money would be fine.

Or you could make some "money" holders that could then hang on their Christmas tree...

I used to buy my kids' elementary school teachers gift certificates to the Scholastic book order. I figured they always end up spending their own money of classroom books.

I was a teacher for many years and my daughter is also a teacher. Teachers often spend their own money on things for their classes, so a gift of money is always appreciated. I received a variety of gifts over time and always appreciated the giving nature of my students. Many of the mugs I still have and use. :)

How about a mug rug for all those mugs? With some cash . . .

I often gave money or office supply gift cards to my son's teachers.
I gave it to them at the end of the day when I picked up my child from school, or just left it on their desk, in a card, when I volunteered in the class.

I don't know why, but other parents seemed to act weird about that sort of thing when they saw me gifting. I enjoyed being anonymous.

When I taught, I would have loved this...

I think you have a great idea. I used to spend thousands of dollars a year of things for my classroom that wasn't in the budget. Teachers always get way too many tote bags, coffee mugs, cutsie stuff that they will never use. I applaud the practical.