I don't know about long term rent, but there are places where you can at least have vacation length rentals of some natural structures. I think they're usually marketed under ecotourism.
Wheaton Labs actually has some examples and the opportunities to see and practice some of the different building techniques. There are a variety of different approaches used on this property and they are still actively developing it.
The GAPPER program
https://permies.com/t/46350/gapper-program sounds like it is closer to a work/study program that can give many opportunities to build experience in different techniques. The first post in that thread also links to the SEPPers program which gives you a little more control over your lodging and a lot more freedom in your time and activities. My family took part in that this summer and it was a great part of our vacation.
It's the one one I have experience in, but hopefully there are other people who can link to other sites with tours, rentals, workshops or related activities that are relevant to your needs. I do want to note that many natural building techniques are highly influenced by their location. My area of the country will probably never produce a WOFATI, for instance. We just don't have the woodlands, or deep enough soil before you have to start blasting bedrock.