How are lipid-coated mRNA-based vaccines transported into cells for expression?
I can't give an authoritative answer on this because my PhD work was based on mRNA delivery using peptides instead of lipids, but many of the concepts are the same. I also don't have time to provide proper citations, but this is probably too much for a comment.

Anyway, mRNA is a very fragile molecule, it can be broken down extremely quickly in the blood by serum nucleases. Therefore we must protect the mRNA by mixing it with some other chemical, in this case a mixture of positively charged lipids. The positively charged lipids are attracted to the negatively charged phosphate backbone of the mRNA, and form a nanoparticle trapping the mRNA inside and hiding it from the nucleases. The lipids are often, but not always, PEGylated, which means that a molecule of polyethylene glycol (PEG) is attached to the lipid. The PEG groups form a layer on the outer surface of the nanoparticle that helps prevent protein binding.

These mRNA lipid nanoparticles are then injected into a patient. This will probably be an intramuscular injection so that the particles are most likely to be taken up by muscle cells. The exact mechanism explaining why lipid nanoparticles are attracted to cells isn't entirely clear, but it's likely a combination of electrostatic attraction between the positively charged nanoparticle and the negatively charged cell membrane and proteins that bind to the nanoparticle and are then recognized by receptors on the cell surface.

After the particle is taken up by a cell it is usually trapped inside a membrane-bound structure called an endosome. As the endosome matures it is acidified, and as the pH drops from about 7 to about 5.5 the nanoparticle is disrupted. The lipids that made up the nanoparticle can merge with the lipids that make up the endosomal membrane, disrupting that membrane and breaking open the endosome, allowing the mRNA to escape into the cytoplasm.

Once in the cytoplasm the mRNA will find a ribosome and produce protein. Just like you said, the protein in this case will be membrane bound and expressed on the outer surface of the cells for eventual immune recognition.