So in that case the mRNA is more like printer ink than like a blueprint? Or both at the same time? Is it possible for a cell to still produce the protein when it runs out of mRNA?
It's a print _job_. The cell is printing more bits of cell all the time, we just slip in some extra messages that print foreign proteins. The immune system identifies the proteins as foreign and produces antibodies to destroy them, which primes it to destroy matching COVID viruses.
mRNA is "messaging" RNA. It sends the message to the protein factory to make more of a protein structure defined by the DNA, just like a print job sends the message to the printer.
I'm not an expert in this but... If DNA is like a blueprint, mRNA is like an email sent from the architect to the contractor to instruct them to carry out specific work.
If DNA is a blueprint, mRNA is a photocopy that's used as a working memory of it for a ribosome to use while making a protein. It's chemically unstable and soon falls apart. (The individual nucleotides night be recycled into other mRNA strands that the cells make.) Since the vaccine mRNA does not have any corresponding DNA to make more copies from, that's the end of it.