I am corresponding via email with an industry researcher regarding an internship. He always signs off his emails with his first name, Bill. Still, I have addressed him as "Dr. X" for several emails now. I realize that if I am offered this internship I wouldn't want to keep calling him this and would prefer to be on more casual terms. Should I just abruptly switch to his first name, or wait for him to actually tell me to use his first name?
Since he is academically your superior and you are trying to get a research position, I'd continue addressing him as Dr. X. I'd continue doing this until he directs you otherwise.
Now, if you get the internship I would continue addressing him as Dr. In person, at least initially. Eventually you may notice other people who work under him that call him by Bill. That would be a good time to say "I notice everyone else addresses you as Bill, would you prefer I address you as Bill or Dr. X.?"
Either way, don't get in a hurry to be on a first name basis.
Easiest solution, ask him.
Dear Dr. X,
[Rest of the email]
P.S.: do you prefer me to call you Dr X, or Bill?
No mistake possible.