There is no evidence that the mRNA vaccines alter DNA, indeed according to the science of cell biology, such a thing is just shy of impossible for each cell. To happen on a large scale IS impossible.

Even if a single cell did have its DNA altered by the mRNA in a vaccine it would almost certainly not be imbedding in an area where it will have all the other DNA elements required to actually be produced as a protein (promoter regions, enhancer regions, proper reading frame, etc.).

Even if that single cell DID have it imbedded with all the regions required it would almost certainly not survive long because the protein is designed to produce an autoimmune response killing the cell.

Even if the cell did survive it would not be likely to be a blood cell.

Even if it was a blood cell it would not be likely to survive long enough to be transfused, nor be one of the cells in the transfusion.

Even if it was transfused it would almost certainly be killed by the recipients immune system because the protein is designed to produce an autoimmune response killing the cell

Even if it did survive it could not then infect the recipients cells in any way. Its a surface protein, on a blood cell, that's not what they do...

Even if it did infect one of the recipients cells (cells don't infect other cells, but I'm still playing along) HOW COULD IT THEN CAUSE DNA MODIFICATION IN THAT NEW CELL!!???!!!???!!??

TL;DR: no.