DNA vaccines and in this case mRNA vaccines would likely be taken up by myocytes (muscle cells of the arm). While not the optimal antigen presenting cells (APC), they have the proper co-stimulatory molecules on their surface to present the vaccine antigens to T cells. More potent APCs would be interstitial dendritic cells and these would also be present, albeit to lower levels then muscle cells. Unless there is a secretion signal sequence encoded as part of the expressed spike protein for extracellular transport, the majority of the vaccine would be expressed intracellularly and loaded into Class I MHC for cellular CD8 T cell responses, but some vaccine antigen can and will be dumped extracellularly which would elicit the antibody response. There may be some microtissue trauma from needle injection which causes some minor inflammation, aiding in recruitment of immune cells.
mRNA degrades pretty quickly even during "normal" bodily gene expression, but I think I recall hearing this vax is delivered in a liposome which provides enhanced protection against degradation.