What type of Engineering is associated with the roller coaster industry?

That is an interesting question. There will be a lot of engineers that support the design and installation of a rollercoaster;

ME with the car and rail designs, structural engineers for obvious reasons, electrical and software for the controls, safety engineers, manufacturing engineers for making the parts and cars, construction engineers during the build, etc. The over architect of the roller coaster may be a physicist.

I am not that familiar so I sought information online at this site:

https://study.com/articles/Roller_Coaster_Engineer_Job_Description_and_Education_Requirements.html

No specific major exists in roller coaster engineering, though a design team is typically made up of structural, electrical and mechanical engineers. Many universities offer bachelor's degree engineering programs in these disciplines. Students interested in designing roller coasters should consider including additional physics, mathematics and drafting courses to their curricula.

The main responsibilities of a roller coaster engineer are deciding where the tracks should be placed, picking the right roller coaster for a particular environment and making sure all aspects are safe with each roller coaster. To make sure rides are safe, these engineers test them out, making sure all of the weights and speeds are set correctly. Aspiring engineers often need a license, though requirements vary by state.

I like what Mr. Arche replied with. Most projects take a number of engineering and technical fields as he indicated. You could be any one of these specialties, or a project manager that manages the team/project. Good luck.