The payload of many viruses is RNA very similar to mRNA and injecting it into cells is how they reproduce.

Given how frequent are mutations in viruses, if there was to be a zombie apocalypse, natural selection would have done it by now.

Some kind of deadly pandemic is more likely to happen due to mRNA manipulation (and it would have to be on purpose, as the mRNA would need to come with instruction to reproduce in a way that lets it travel from cell to cell, i.e. de facto be an RNA virus which hardly happens by chance) than one which produces zombies.