In general yes. You have the same bus for the entire trip. On the best of Switzerland one day the driver did go home for the night. I believe the guide advised us to grab all our stuff.
Almost always the bus driver has deep storage. This is for stuff you will not need for an extended period or until the end of the tour. Deep storage is underneath the bus in the luggage hold. One driver had a plywood crate built for deep storage.
There is an overhead shelf on the bus for storage during the day. There’s generally enough room that everyone that wants one can get their own seat. So the seat next to you can be used as a place to leave reading materials jacket etc.
You can ask your guide at the orientation meeting if you will have the same bus for the entire tour.
Have a great trip. Be careful RS tours are addictive!