If Yamato does end up joining, I can see her taking on the position of that of an Apprentice.
As far as we know, Yamato has been trapped on Onigashima for the last 20 years since she was 8. This doesn’t give her a whole lot of options in terms of learning new skills and figuring out what she really wants to do with her life. I think the whole “wanting to be Oden” thing is just her way of compensating for that and is kind of an escape for her.
With that being said, I still believe that she will continue to see Oden as a role model but will no longer try to literally BE him. I believe she will be inspired by Oden’s adventures so that she can find her place in the world. What better way to learn new skill than as an Apprentice under the Straw Hats. Her gag would probably be her trying to mimic her crewmates’ mannerisms like she used to do with Oden. And it makes for a bit of a soft parallel with her father as Kaido himself was an Apprentice under the Rocks Pirates.
Here is why I don’t believe the hypothetical Apprentice position will be held by either Momo or Tama. The two people Luffy is directly influenced by and looks up to the most are Shanks and Ace. Both of them rejected a child from boarding their ship, Shanks with young Luffy and Ace with Tama respectively. Because of this, I don’t believe Luffy will act any differently. Plus the fights in the new world are only going to get harder and the SHs won’t have the time or energy to be babysitting 8 year olds.
With that being said, I do believe if the Apprentice position ever ends up being filled at some point, Yamato seems like the strongest candidate. This is because she is proven to being a capable fighter who will easily be able to hold her own against strong new world opponents.