GMOs can definitely reproduce... Not all of them can, but most do. One problem you might run into with saving seeds from grocery store chiles is, some producer irradiate their pods so the seeds won't be viable...
To answer your question more directly though, yes, some commonly available grocery peppers are GMO. But, you will likely find that the seeds are good anyway. If you're looking to avoid GMO chiles, best bet is to buy seeds from a reputable seed vendor. If you're only worried about whether or not the grocery's pods will germinate or not though, my best advice is to sow some seeds and see if they work. You don't need to dry the seeds out first; you can go straight from the pods to the dirt to check and see if the seeds are good.
Hth. I wish I could tell you which chiles are most likely to be GMO or not, but with our current laws, there isn't much transparency. I know Bell Peppers from the supermarket are likely to be GMO. I suspect a lot of these Jalapeños these days are, too...