I have not been to Beppu, but in general:
There are “route buses” (路線バス) and highway/tourist buses (高速バス・観光バス). The highway buses/tourist buses generally have luggage space underneath and steps up into the cabin. While ones with lifts do exist, they are VERY rare. So, you would need a folding wheelchair to put with the luggage and then be able to climb the steps.
The route buses are general buses used by residents for normal transportation. In big cities nearly all of them are “non-step” accessible buses with a small ramp the driver can put down for you.
In smaller cities/the country, these exist but due to budget constraints the roll-out of the new accessible buses is slow. That means they may only be on some routes, or on a route but in a pattern like “every third bus” etc. (There are some older versions of accessible buses that still have two-steps in and they use a steep/long ramp that can be a bit scary!).
I’m guessing that the reason the bus is asking you to make a reservation is because not every bus is accessible and they could accommodate you if they know your schedule. ie you say you want to get the 11:30 bus from X station on a certain date and they can adjust the bus schedule so that bus is one of the accessible ones.
The bus company likely only has Japanese-speaking staff so you may want to ask your hotel or tour guide to call. Alternatively, you could look into hiring an accessible taxi if you are concerned about the buses.
Hope this helps!