You do need a special license to run a day care service, and you should need a special license to drive people around unless you have a few million in the bank to pay for the damage you cause.
Lots of people work in the underground economy to avoid taxes, and while there is some short term gain to be had be outside the system, there are reasons why the system exists. Some of it bullshit, like wars and NSA and so forth, but some of it comes out of the labor movement and is designed to help and protect workers. Things like unemployment and workers comp. By working under the table, when something goes wrong, you are really screwed. And big business is always looking for ways to shift the costs of doing business onto the worker. This is probably one of those ways.
I don't know about every state, but one of the big games businesses try to play is telling people to become independent contractors. They think that if their workers are ICs, they won't have to pay workers comp premiums. Except the WA state statute doesn't talk about "employees" -- it talks about "workers where the essence of the contract is personal labor." So a while back, it was a popular way for taxi companies to shirk their responsibility by leasing cabs to drivers and making them independent contractors. Didn't work and they got spanked because the drivers provided only personal labor.
In the case of this company, where they act as dispatcher arranging payment, pick up, drop off and act as boss (they'll essentially fire you if you don't live up to their standards) -- that's personal labor. And while you may provide your own car, that isn't good enough to get beyond the "worker" definition (been tried). So anyway, if this company is operating in WA and not paying premiums, it's going to get fined, and if a worker gets hurt while driving, they'll be on the hook for all the claim costs.