These are not mystery shops at all. The news organization is Reuters, which is working with Ipsos, a research organization, to collect the data. Headway is the company that's handling the hiring, just as they do for other companies and other types of positions.
I'm not an election agent for them, but a bit higher up the food chain. Won't mention the state here, even though it's not the one I live in.
They have been adding agents like crazy. Each agent is supposed to go through a training and upload mock data via smartphone to make sure they know how to do it correctly.
It's true that in some areas the vote counts are done quickly and posted online quickly, but keep in mind that problems reporting the results can and do happen. For a news agency like Reuters, having folks out in the field is good insurance in case a city or town's system runs into electronic or other problems.