The mRNA must enter a cell to express proteins. The delivery vehicle is a "lipid nano-particle", a particle that can contain a payload and attracts surfactants to protect the payload but also allow it to penetrate cell walls. They're a bit unstable, thus the need for refrigeration. The other way this can work is using a virus, which is what the Oxford vaccine does.

You can do protein vaccines, and some are coming, but they're slower to develop as you have to figure out a way to manufacture them; with a nucleic acid you can have the body do that part for you. Additionally, the mRNA produces a more complete response; some parts of the immune system react to proteins expressed on the surface of cells, which direct protein exposure doesn't do.