The short answer is "It's not possible". The longer answer is below.
All but a handful of the genes in your genome are contained in the nucleus that resides within the cell. (Those other handful of genes are in the mitochondria, but that's a whole separate discussion). Envision the nucleus as a soccer-ball, but a soccer ball that is covered in tiny passages that connect the outside of the ball to the inside of the ball. Those passages are nuclear pores, and the set of proteins that create those passages are called the "nuclear pore complex", or NPC for short.
There are two ways for a molecule to be transported across the NPC. (This process is called both "nuclear import" and "nuclear localization"). The first method by which nuclear import can happen is "passive diffusion". In this process, if a molecule is sufficiently minuscule, random motion (technically called "Brownian motion") can cause the molecule to simply "jiggle" its way through the NPC. It should be emphasized that in order for passive diffusion across the NPC to occur, the molecule has to be REALLY small, with a practical mass limit of about 60 to 80 kDa (kilodaltons). The absolute upper molecular mass limit for passive diffusion across the NPC is around 120 kDa.
Each of the mRNA oligos in the the two mRNA vaccines has a mass of about 1400 kDa. There's no possibility of nuclear import via passive diffusion.
Passive diffusion isn't the only (or even the main) mechanism by which molecular nuclear import occurs. The primary mechanism by which molecules are imported from the cytoplasm into the nucleus is via a process called the Ran-GTP cycle that utilizes enzymes known as "importins". Importin enzymes serve an incredibly important function in cellular processes because they act as a "cargo truck" for mediating the passage of other proteins across the NPC and into the cell nucleus. There are all kinds of complexities involved in Ran-GTP (and if you're biochem nerd, it's fascinating how it works), but the basic gist is this: importin gets exported out of the nucleus, it bumps into a protein that contains a "nuclear localization signal" (otherwise known as a NLS), binds that protein to itself, mediates that proteins passage across the NPC into the nucleus, unbinds from the protein, and starts the cycle over again. Genuinely, the best mental model for understanding that process is a cargo truck puttering its way out of the nucleus, hitching up the first specially-marked protein it bumps into, hauling it back into the nucleus, dropping off the load, and heading out again.
Here's the rub: how does that importin enzyme know which molecules it can pick up and haul back into the nucleus? If you remember any of your high school biology, you may recall that a protein is made up of a long string of amino acids that are folded up in an incredibly complex way. (And peptides are much, much shorter strings of amino acids, without all of the complex folding). An NLS (nuclear localization signal) is a special sequence of amino acids within a protein or peptide that mark it as eligible for nuclear import. (i.e., those NLS sequences are what importin binds to). Without an NLS, no protein or peptide (that is too big for passive diffusion) can be translocated across the nuclear pore complex.
All DNA and mRNA molecules are nucleic acids formed from nucleotides (not from amino acids) and therefore lack an NLS. The protein product of the S-gene in the vaccine's mRNA oligo (like the S-gene in the SARS-CoV-2 virus itself) lacks an NLS. Without an NLS, there is no way for nucleic acids (including mRNA) to participate in the Ran-GTP cycle that imports proteins into the cell nucleus.
I won't go into the details here, but I should also point out that even if you could "magic" the mRNA of the vaccine (or the SARS-CoV-2 virus itself) into the cell nucleus, it still wouldn't integrate into the nuclear DNA. That mechanism would require an enzyme known as a reverse transcriptase, which is not encoded within either the vaccine nor the SARS-CoV-2 mRNA.
As an aside, HIV (another RNA virus) is as wicked as it is because it does encode proteins which contain an NLS, and it does encode the reverse transcriptase (and other reverse-transcription factors) neccesary to edit its genome into our cell's nuclear DNA. These types of viruses (called "retroviruses") have been around longer than land-animals, and a pretty big chunk of non-coding DNA in mammals is made of up viral DNA from retroviruses that infected our common ancestors in the unimaginably distant past.