I’ll admit the writers are entitled to fudge reality, especially when it comes to legal technicalities. It’s not a procedural or legal show, plus there’s a lot of variation about how states handle things between states, and even in the same one over time.
Anyway, the whole adoption still was shady. Regardless of whether Regina filed magic paperwork, what she filed was based on a sham world she created. We know there were no follow-up state workers to do home visits. Her references were likely people in her curse, programmed to say nice things about her (because who dares to do otherwise with Madame Mayor?).
The agent didn’t come across as some greedy, child-seller. He seemed more like your standard agency shill, someone who punches in their time but doesn’t try too hard beyond the minimum required of his job. So I guess it works that he wasn’t curious about Regina. She had the paperwork, she looked the part, and then I guess we’re to assume he punts the ball to the state government to do the follow-up. He files his report with them, and then some clerk gets a magic report that all is well and no one is the wiser.
Somehow Regina was able to get Sydney to talk to people to get information for her, so there’s some influence from SB to people outside of it, though I don’t know if it is the same as magic spells. It might be that money /credit created with magic in SB can affect databases outside of it or something. I’m still not sure how that works with Mm’s credit card that Henry stole.
Might be time to find our “pot holes” thread.
“If I had a world of my own, everything would be nonsense. Nothing would be what it is, because everything would be what it isn't. And contrary wise, what is, it wouldn't be. And what it wouldn't be, it would. You see?” -- Lewis Carroll, Alice's Adventures in Wonderland & Through the Looking-Glass