Don't make it harder than it is!
I say as I write a long and complex answer lol.
Your concern is the vanity has a convenience plug which is not GFCI protected.
However, here's a secret about GFCI protection: It is a zone of protection not a receptacle.
You're accustomed to seeing a particular implementation of GFCI that contains a combo GFCI protection device + 2 sockets. Well of course it puts its own sockets in its protected zone, but it can protect anything else also. That's what the "LOAD" terminals are for. It is the only legitimate use of LOAD. Don't use it accidentally.
However -- you can absolutely use the downline protection feature deliberately.
The electricity going to this vanity must come from somewhere. Since you've already worked on it, you know which circuit breaker it is. So turn off that breaker and survey the house for what other points-of-use have gone dead.
Now guess at the serial route from point to point, and try to figure out one that is between the vanity and the electrical service panel (source). Strongly prefer a receptacle over a switch (still possible with a switch if power comes to the switch and not the light). Then open that up, photograph how the wiring is now, and then disconnect the downline hot wire... insulate it with electrical tape in case it's actually the supply hot wire (we don't actually care at this point)... then power the circuit back up to see if that cut power to the vanity. If it did, this is a good point to install GFCI *protection).
If it's a receptacle, it's a routine installation of a GFCI receptacle. Hook up the suspected supply hot+neutral only to the LINE terminals only, plug a lamp into the socket, go back and turn the breaker back on, and see if the lamp powers up. If it does, you have the right wires on LINE. Power back down, hook the others to LOAD and the vanity should now be GFCI protected. Trip the GFCI with the "Test" button and confirm the vanity loses power.
If you "accidentally" protect a large part of the circuit, shucks o-golly LOL. Make sure to label all receptacle outlets "GFCI Protected" and "No Equipment Ground" if applicable. If you hate the blue stickers feel free to use any marking method that is not hand-written. I like white cover plates + a P-touch or Brother label maker. Or if you know a bro with a laser etcher, go for it.
If it's a switch, that's a bit trickier, but still possible if the supply wires come to the switch first, i.e. there would be 3 hots and 3 neutrals inside the switch box, with all 3 neutrals spliced to each other. Because they make GFCI+switch combo devices too. Ask if you have that situation.
If the vanity is the very first point on the circuit, we'll have to resort to a GFCI circuit breaker (yeah, those too).