On Friday, after another semi-random search through a trail of links on various government websites, I discovered a portal specially designed to streamline this process. I applied for my L number (state tax ID) and actually got a prompt reply!
I received an email this morning informing me that my application had been accepted and that my number had been approved. However, they didn't tell me what my number actually was, and apparently I have to wait five business days for it to be searchable on the Department of Assessments and Taxation's website. I can only hope they're also sending a paper copy by mail. Maybe.
Next, we apply with the county.