No, not at all. First, let's look at the basic structure of a human cell:
So inside the cell, you have the cytoplasm (a sort of gel that allows things to move that need to but otherwise helps hold structures in place), a bunch of organelles including ribosomes, and the cell nucleus. The nucleus is where your DNA is and there's a membrane around it to protect it. The only way in or out is through the nuclear pores.
So all day every day, parts of your DNA are used to create proteins your cell needs to live and do its job. DNA doesn't leave the nucleus, so an enzyme inside the nucleus unwinds part of your DNA and transcribes the needed piece onto messenger RNA which gets constructed in a process called transcription. Once that messenger RNA strand is completed, transport proteins bring it out of the nucleus and into the cytoplasm. One of the organelles in the cytoplasm is something called a ribosome. Its only job is to take messenger RNA and read it to produce whatever amino acid or protein is encoded. So your DNA gets transcribed onto a messenger RNA strand and that mRNA strand is used by ribosomes to create the proteins necessary for life. This happens in every cell in your body all the time.
Now if a virus (like SARS-CoV-2, the virus responsible for COVID-19) infects one of your cells, it injects its RNA into the cytoplasm and uses your ribosomes to create complete copies of it. There is no regulation on this; this replication continues until the cell bursts apart with new viruses, each of which can then infect new cells.
With the mRNA vaccines, a strand of mRNA which encodes for one piece of the virus enters some of your body's cells and is read by ribosomes to produce a surface protein from the SARS-CoV-2 virus. That surface protein is just one small part of the virus and can't infect anything or replicate itself, but it's recognized by the immune system as foreign so it triggers a normal immune response. The ribosomes are just doing what they always do: they got some mRNA and they produce the encoded protein. The mRNA itself breaks down rapidly, with a half-life of just a few hours. That means all the mRNA in the vaccine is gone in just a few days. This is in contrast to the self-replicating SARS-CoV-2 virus. It doesn't break down inside the body. It continues infecting cells and replicating until either you are dead or your immune system has destroyed every last virion.
So the important points here are that the mRNA vaccines don't do anything with DNA. Their mRNA has no way into the cell nucleus where the DNA is. If it magically popped in, what happens to mRNA inside the nucleus? Those transport proteins just bring it out through a nuclear pore. And it wouldn't matter anyway because transcription writes DNA to mRNA; not the other way around. mRNA has no way to unwind DNA, no way to make cuts or inserts, and no way to repair DNA. The mRNA just enters the cell cytoplasm and causes some proteins to be created which then train the immune system on how to recognize and destroy the SARS-CoV-2 virus.
I hope this helps your understanding of how this all works.