I recently came across the following article attempting to answer that question. You can click the link [ here ].
The following is a response by Hyohyun “Esther” Park. Esther has a BS in Biology and several years of experience doing genetic research.

What the article says, that the mRNA won’t affect human DNA, is correct as far as I know (and learned from schools).
mRNA is one of the 3 RNAs our body uses: mRNA (messenger, to copy DNA sequence and be read by ribosomes to make protein), tRNA (transfer, to connect/match mRNA and amino acids of proteins), rRNA (ribosomal, to form small ribosomal proteins). Since mRNA is just a copy of part of our DNA sequence, it will not affect our genome but only affect the proteins our body makes.
My previous company (Orbis) was trying to enhance the success rate of the mRNA vaccine delivery system because it has been so hard to generate enough antibodies after inoculations. If I had a choice, I would probably get the Pfizer vaccine because it’s the most effective/reliable one. I have heard some stories about Moderna that they were never successful in manufacturing anything before this C19 vaccines… but that’s just me. Other companies’ vaccines are made of viral vectors, which are not as effective as mRNA (as far as I know).
I would also make sure that the vaccine I get is properly transported and stored until the inoculation (if there is any temperature requirement) since temperature regulation could play a significantly important role when it comes to medicine. And that is why I trust Pfizer one the most since it must be stored/transferred at -70C probably in liquid nitrogen tanks (which I used when shipping the cancer vaccines that I made). Such low temp prevents the degradation of the product the most.
To clarify what I said above at the end of the 3 types of RNAs, “Since mRNA is just a copy of part of our DNA sequence, it won’t affect our genome but only affect the proteins our body makes,” the mRNA I mentioned here was “human mRNA” in our normal human physiology.
The mRNA vaccines will contain the “viral sequence” to create enough antigens (in this case, the viral protein) in our body. Our DNA stays intact the whole time, it is just the additionally made antigens of viral proteins for our immune system to learn to react to.
The above information is offered only as personal opinion and is intended to be a recommendation. The reader should consult their personal physician to make a responsible decision regarding vaccines.

The mRNA C19 vaccines already have the mRNA in them, hence there’s no need for transcription in the human cells when the mRNA vaccine is injected. So there won’t be any R-loops formed in our body (R-loops form during transcription), which means no “m6a” created after inoculation. The m6a part was my main concern but it won’t even be made with the vaccine, so I think there’s no need to worry about it. The mRNA C19 vaccines already have the mRNA in them, hence there’s no need for transcription in the human cells when the mRNA vaccine is injected. So there won’t be any R-loops formed in our body (R-loops form during transcription), which means no “m6a” created after inoculation. The m6a part was my main concern but it won’t even be made with the vaccine, so I think there’s no need to worry about it.
Esther update ~ Feb 13, 2021