I logged onto my Seattle and WA business license pages and everything says annual returns. I was under the impression this job required quarterly tax payments...am I mistaken? Am I thinking Federal returns?
For federal taxes, as far as real world stuff goes, you can also just keep it simple by ballparking the math, or ignore doing quarterly payments at all.
Keep it simple could be seeing your gross income for a quarter and sending in an estimated payment that would be say 15-20% of gross income, not taking into account deductions or anything else. This could/should be close to what you might owe, but just all ballpark figures to keep it easy. Then you do the full on proper math once a year at the end of the year along with whatever other tax stuff you may have going on. So long as you're within a fairly good sized margin of error when you file at the end of the year you have no penalties.
Alternatively you can skip quarterly payments altogether, and then just pay it all lump sum at the end of the year. It is NOT illegal, or unusual to do it this way. There is a small penalty for doing this, but many businesses do this every year on purpose because they can put the money to better use throughout the year.
ALSO keep in mind that if you have a regular job, and usually get a refund, you are overpaying there, and hence might need to send in less to cover rideshare income. It all depends on your particular circumstances of course, and you should definitely read up on it all before doing anything crazy or whatever.