Sizzix dies will cut through some fabrics. I've cut fabric with mine, but sometime there were a few threads or more that didn't cut. Those could be cut with scissors.
I know there are other machines that do it.
Now that I think of it, Downrightart posted a thread about something she got that cuts quilting squares out of fabric (and other shapes). I know it works better than my little Sizzix.
I'll try to find the thread for you.
Oh! And welcome to the forum!!!