You probably noticed that vinyl floors in a camper are installed before the furniture and cabinets.
With this being said, I remember doing a similar job in the past.
First, I removed all the moldings everywhere on the floor (don't know if you have any in your camper). I did leave the old vinyl there.
Second, Since the new vinyl is wider and longer before you make the cuts, I took all the floor measurements in the camper (that is the tricky part) and transfer these measurement on the new floor. To make things easier, you can use some large sheets of wrapping paper, and cut them following the cabinets and furniture.
Third, I cut the vinyl to those measurements and brought it in the camper. Up to you if you want to use an adhesive.
Fourth, I install the moldings. If I remember correctly, I bought new moldings and stained them.