Not every thread is going to go through your machine, so you're probably stuck with regular mercerized cotton, which should work fine with the right needle. If you feel insecure, you can stitch around the patch with a leather needle and NO thread to make holes around the edge, then use a large hand sewing needle and carpet thread and stitch the patch in by hand.

Thank you Theturtlelady for your reply. I was thinking that I could do that but I have to wait until my order from JoANN's comes in. I ordered a leather needle from them but it hasn't arrived yet. I could also use glue but I don't know if it would hold up since leather uses rubber cement when glued to other leather, I don't know if it would work on fabric.

You could use a little glue to hold it in place while you stitch, but the tapestry fabric will absorb it and get crusty and I don't think it would hold long for the use you have in mind without sewing.

I altered leather jackets and pants on regular home machine with regular thicker needle. I always double my thread on the bobbin and on top. I like to put strip of paper on top of the leather when I sew, it glides nicely, ,long stitch works better. Don't know how your machine will take it, maybe you will have to pull it a little bit as you sew.
Go slow, the thread on top could bunch up so watch it and even it out as you go.