I would just ask them if they would have time to sit down for lunch sometime. Say that you've heard good things about them and their practice (assuming you have), and that you would love to pick their brain for ideas that have led to their success. People do this all the time in the business world.
The hard part is getting through the front desk to talk with the dentist. One of the functions of the front desk staff is to protect the dentist's time. Here's a few ideas for this:
Call and ask to shadow for just the morning or the afternoon. That way you can get to meet the dentist one-on-one and ask them to lunch yourself.
See if the website has a 'contact-us' email. The front desk will most likely forward the email to the dentist directly and they'd respond within a few days.
The only reason I would hesitate to call and ask the FD staff for a lunch appointment directly is the possibility that they'd say, "Uhh, I don't know, I'd need to ask Dr. Dentist, but he's busy right now." Then you'd leave a message which might not get to the dentist in its entirety or in a timely matter.
It's always best if you have a connection with someone in the practice. But sometimes you have to just go for it!
