Individual cell DNA gets altered all the time. That's how viruses reproduce. Our bodies sometimes even do it on accident (cancer). On a completed organism, it is impossible to do it in any meaningful way without significantly harming the organism. So the idea is downright silly, unless you're suggesting they are simply trying to kill us, which would be more easily accomplished with poison or just about any other way, so it's still silly.
I'm still open to other conspiracy theories (such as using vaccines as a covert method of microchipping, or as a means to reduce fertility)
I'm also open to arguing they are rushing to distribution without adequate testing. But altering our DNA is just not plausible.
(If we could, the ethical issues she brought up would be valid)