https://www.ilsos.gov/PickAPlateWeb/index1.html
You can type in plates here to see if they're taken. "SOXFAN 56" wouldn't work because it can only be a total of 7 letters/numbers... I think?
I have "CHSOX 56" with White Sox license plate holders and -- no joke -- got pulled over by a cop on the North side who told me (jokingly), "You can't have those plates around here," and then drove away. Never happened again. Once leaving the O'Hare parking lot, the cashier asked me if I was a player and I said, "No, just a fan." Other than that, it hasn't been noticed very often.
If you walk around the Sox parking lots, you can see a lot of personalized plates from Sox fans. Of course, that doesn't help you now.