The yellow pages are a simple way to find the DPMs in your area. It should sort physicians by specialty.
As for how long you'd need to visit the office for, I'd say about a week (5 days approx 9am-4pm) is standard. Try to go for at least a few days of office visits (clinic type patients) and then maybe ask if you can visit for that podiatrist's OR day to see a few surgeries. It'd be pretty standard to request a letter after that. The letter from the DPM isn't so much of a recommendation and isn't expected to be one (they barely know you and don't know your academic prowess). The DPM letter is more just to state that you've experienced the field of podiatry firsthand and maybe comment on how interested or professional you appeared during that shadowing.
Maybe I'm just misundersanding the tone of your post, but your goal shouldn't be to do the minimum. If you are interested in the profession, you should enjoy shadowing and maybe continue to stop into the office periodically as permitted even after you get the letter. I sent a thank you to the DPM I shadowed and left a box of chocolates for his office staff on the last day I was there. That might be viewed as overkill, but I did appreciate him letting me tag along and I learned a ton (he scheduled my visits wisely to show me many different aspects of the profession: C&C, orthotics, OR, minor office surgeries, etc). I have since emailed him occasionally to let him know how school is going and stopped back to visit for a day during holiday break. It's always good to learn as much as possible and develop relationships; podiatry is a fairly small community and word travels fast.
just another option if you have not tried that yet. 
