Hairspray removes ink from fabric, but I don't know what it might do to the finish on the doll faces. If you try hairspray (on a Q-tip or cotton ball), I'd try it along the hairline to make sure it doesn't discolor the skin.
Good luck!
I have some vintage (like anything from the 1960s can be vintage

)
Anyway, in some of my first attempts to clean indeterminant ink stains, I tried alcohol. All it did was made the ink smear and there was a larger stain than before.
I'd suggest the paste of baking soda. It's gentle and at least won't hurt anything if it doesn't take the ink off. I don't know if they still have it, but AG had a 'doll cleaning kit' you could buy. It was actually a small bottle of baking soda packaged with an AG white washcloth.
I would try baking soda first. If that doesn't work and there are a lot of ink spots, I'd suggest sending her in to the doll hospital.
If Another thing I have used on vintage dolls that works and usually doesn't seem to hurt is pimple cream with Benzoil Peroxide. The Peroxide works as a mild bleach, so make sure to try this in a 'non-exposed' place first.
You put a paste of it on the area with a Q tip. Be careful to get it only on the ink area and not on clothing, hair or anything else since it is a bleach. Then put the doll out in the sun, which will accelerate the bleaching process. Use paper towels or something to cover other parts of the doll and expose only the stains.
Whatever you try, if you do send the doll in to AG, be sure to tell them all the things you tried and in what order. If they are going to try to remove the stains, they may need to do additional steps depending on what you did.