Let me get this straight. You haven't started college, and your plan is:

1) Double major in accountancy and psych. No problem here.

2) Get into a fairly competitive internship (accounting), then an internship for clinical psych. The first is something you do in the hopes you'll get a job offer to make the big bucks once you've completed your CPA. While you would get work experience, you would not get the main benefit, a job, because you do not plan to enter the workforce afterwards. To be frank I'm not even sure what the goal of a clinical psych internship would be. I'm sure you'd learn a bunch but it wouldn't pay well. How are you financing all this again?

3) All that "whilst" getting into med school. First off, why whilst? Genuinely curious. Honestly, doing internships over the summer while in undergrad isn't too crazy, although double majoring with a major that - assuming you actually want to use it - requires a shit ton of credits like accounting makes it a lot tougher.

4) Here's where I don't really get what your scheme is. After you get into med school, you want to work as an accountant for two years while getting a master's? First off, even without med school in the picture, no. You're not doing that. I know accountants. It's not a 9-5 job at most places, more like 7 AM-6 PM during the slower months. Second, what are the med schools you got accepted to doing with your seat during those two years? Cause they're sure as hell not just keeping it warm for you without a good reason.

After all of that time, would it be possible to apply?

In this hypothetical scenario, after all that time you have already gotten into med school and deferred matriculation in order to pursue your dream of being a triple-threat CPA-Psych-MD, so no, you cannot apply, you're already in.

That was all very long, so to answer the heart of your question, no, there's not a time limit. There's a dude pushing 40 in the class above me. That does not make your plan even close to good though.