The RNA based vaccines don't have anything to do with stem cells or dogs.
You may be familiar that cells contain chromosomes made of DNA that serve as encyclopedias/ instructions for carrying out the functions of the cell. To read a particular sentence or paragraph in the book, special enzymes (types of proteins) create RNA versions of the information. Messenger RNAs are RNAs whose information is then translated by other complexes in the cell to make a particular protein. The Pfizer and moderns vaccines are at their heart a stabilized messenger RNA that contains the information to make the SARS-CoV2 spike protein. But, the vaccine RNA has to be able to get into cells, so the companies surrounded the stabilized mRNA with lipid based spheres (nanoparticles). This helps the vaccine to deliver its cargo through the outer layers of our cells, which are composed of phospholipids. The companies wouldn't have to use any cells to make the RNAs or their delivery shells. Their technology is a lot easier to scale than, for example, flu vaccines that use chicken eggs.