Probably depends on state and facility. You could contact your BON for more info.
I had a discussion with a colleague about this a few months ago. The company I work for also has a training facility for CNA/MA/PCT/phlebotomy. We conduct background checks before allowing them to enroll. My understanding is that the only thing that will bar you from enrolling is violent or drug related felony. I think possibly financial stuff like fraud or theft might affect, but again, think its only at the felony level. We had a student during the time this conversation occurred who we let sign up even if she did have a negative criminal background. The issue will arise for her when it comes time to find work. Because we might allow this for the training center, but a potential employer might have more issues with it.
It's a risk to take I guess. You say your conviction was 5 years ago...Depending on the program you enroll in, 3-4 years...That puts you almost at the 10 year mark after the DUI. You might be able to get a pardon or something to remove that from your record by then. I guess you could get a consultation with a lawyer on top of speaking to BON for your state.
Best of luck!