Yea, those teepees were up during Thanksgiving, and yes, it is a private group that travels around the US setting up those sites.
This was their annual "convention" this year, I think for years they did it in other places in Florida, but for the last couple years they've done it in the group camping spot at the Fort. I've never heard of being able to rent teepees at the Fort, and I don't think it will be something coming anytime soon (although it would be an interesting idea).