Right, so. On your first question of what Ty Kora Tek followers actually do - they tend to perform funerary rites, such as preparing the body of the dead and then leading the ceremonies afterwards. TKT isn't really widespread in Sartar and amongst the Heortlings, however the Esrolians to the south definitely have a much larger space for her in their culture, as she is still, ultimately, one of the Earth Goddesses.
So if your player wants to be a follower of Ty Kora Tek, I'd recommend maybe picking an Esrolian character, instead of a Sartarite one.
Now, as for actually being an initiate - Ty Kora Tek, similarly to Asrelia tends to be a cult in which older women get into. However, it's important to note - this is how things usually are. You are making adventurers, you are making Player Characters here, so the usual can be ignored if need be. Why would a young woman be initiated into the cult of Ty Kora Tek? Here's some possible ideas:
* During her Adulthood initiation rites, she chose to stay in the women's hut along with Asrelia and Ty Kora Tek, instead of following Ernalda and leaving for Yelm's court. There she learned things which she really shouldn't at this stage of her life, but it seemed t hat the two Grandmothers had taken a liking to her. So the character follows TKT, because the goddess herself hand-picked her for it. This would have the potential for a lot of interesting roleplaying, as you have a young woman trying to both deny her natural urges, as well as the expectations of her community, while also playing an important role as a spiritual leader, helping communicate with the ancestors and lead the souls of the dead to their rightful place.
* Again, during her initiation rites, she was attacked by an enemy of some kind, probably Chaos or something else, and it scarred her, making her barren. When she returned from her adulthood rites, the women in the clan were worried and horrified at what had happened to this young woman. However, maybe a follower of some of the Malign Earth goddesses (Maran Gor, Babeester Gor, or even Ty Kora Tek herself) said she can still find a place, and led her away, and she then joined the cult of Ty Kora Tek as a way of coping with the realities of being a young woman who is going to be forever unable to bear children, a rather painful fate in the kind of societies you have in Dragon Pass. Lots of potential here if the player wants to explore a more troubled character, since this situation can cause a lot of drama, both internal and external. And drama, at least in RPGs, is good stuff!
*Once again, during her adulthood initiation, maybe one of the many things that the character encountered was a revered ancestor of hers. Perhaps that ancestor, maybe an important great-grandmother from her family, impressed on this woman the direness of the times to come, as the Hero Wars start to pick up steam, and so she offered to join with her, to both guide her, and help her guide the people around her. So this young woman accepted and her soul became the same as her ancestor. She returned as someone who, while in body is young and fertile, in spirit she was old, and thus joined the appropriate cult. Again, this also gives a lot of amazing situations for roleplaying, as you have a character who is both young and old at the same time, yet not actually either entirely. She would probably be seen as both weird, mad and sacred, a common trope in both fiction and in our ancient world societies.
I can probably sit here for a bit and think up a few more scenarios, but this should give you a good idea of why a starting PC of young age would be drawn into the cult for old women who deal with the dead.