I think they do use different planes. The whole trip was kind of surreal. When I booked we were told to check in at the airport and wait at a certain gate. Now, if memory serves me correctly, we had to have our backpacks weighed and I think we had to tell them our weight as well. We (my dd and I) went to wear we thought we were supposed to wait. No one else there, we were outside the terminal. In a bit someone came looking for us as we were in the wrong place. But with so few passengers I guess it was easy to round everyone up.
I don't know which airline does the tours. I don't know if there is information on their website. I'm going to go look.
I'm back. Everything has changed since we took the tour 17 years ago. At that time Richard Marks was still doing the tours, although he didn't the day we went as he wasn't feeling well. He was a resident, and passed away some years back. It does look like Mokulele took over as the official airline for the tours in 2018. So unless they have tiny planes they may be flying the same ones they use for inter-island in. There are smaller airlines that do charters, but then you'd need to book your own tour as the only way to get in is to be a "guest" of a resident.
Here is the article where I was getting this information:
https://www.hawaiicatholicherald.com/2019/05/15/kalaupapa-how-to-get-there-today/