No. Changing someone’s DNA is extremely difficult. This vaccine uses our body to make a small part of the COVID-19 virus (one protein called the spike protein). Unlike other vaccines, the COVID-19 vaccine does NOT use a live or weakened virus, so the vaccine does not and cannot give us COVID-19.
You may have heard that the vaccine uses something called a messenger RNA (mRNA) which sounds like it might do something to our DNA. This mRNA does NOT come near our DNA. Instead, the mRNA makes our body make this spike protein and then the mRNA breaks down and disappears from our body. The protein it makes then triggers our body to make antibodies against it. These antibodies then protect us against COVID-19.
Here is a link to a short video (about 5 minutes) that explains more about the vaccine: https://www.youtube.com/watch?v=the81FQoAUI.