You can become CD, York wants to be one
I think the becoming a CD thing is not as simple as that, it is not comparable to being born one from the 19 families.
Sure, the Gorosei want York to be happy and productive for them, so they would probably give her some priviliges, some slaves to play with, and some form of diplomatic immunity and slap the label of Honorary CD on her.
I am sure they would have done the same for Vegapunk himself if he wanted it when he was on good terms with the WG.
But the CDs, even the Gorosei themselves, would never like, trust or accept her as one of them, they would only see her as an asset, and I believe the same applies for any Holy Knight recruit who is not born a CD.