I use an older (at least 12 years old) Epson ink jet printer. I have been buying quilt label fabric sheets but notice some are too thin to go through the printer properly. So, I use a glue stick and glue a sheet of card stock to the back of the label (or other fabric). Make sure you have just over 1/4 inch of card stock showing at the top so the printer rollers have something to grab to. And adjust your design position accordingly. Also, don't glue the entire sheet, just around the edges and maybe one in the middle. I find this works really well.
Another thing to consider is that fabric label sheets are treated to hold the colour. I am currently making my own sheets using Bubble Jet Set. I'm thinking that soda ash might work just as well but haven't tried it yet.
You may also want to play with your colour saturation settings on any design as printing on fabric tends to mute the colours. I have never bothered. Do a couple of test runs on paper first.
I wonder if any of the newer printers might actually have a setting for fabric labels like they have for different qualities of photo paper. Otherwise, use the heavy stock setting.
From what I am understanding Epson uses archival ink still, which has a longer permanency. This probably only matters if you want someone to be able to see the printed photo 150 years from now.
Any ink jet printer will probably be fine.
Have fun, and show us what you print.