If I wanted to I could lock up 5-6 contracts a month just mailing about 1000 leads by giving the person their price and seeing if I can unload it but I do not do that.

This move out statement needs to be mailed to tenant within the proper time period for your state, so check landlord tenant laws for your state.

I was wondering if anyone knows a company that will receive the bills, email the tenants what their portion is, pay the bill for me and collect the payment from the tenants, complete with sending reminders, charging late fees, etc. 

Now, when you can pay an online service $30 or so for the same credit/background screening, e-mail applications to interested parties and schedule actual showings on specific days that fit your schedule, free advertising on postlets and other sites, screening tenants is no longer worth the fees he is being asked to pay, and he should rightly be looking for alternatives.   

CloudSee below in a state I work in Oregon.. again the only one that counts in this matter is the department of real estate in the state your working.. so I get a e mail flyer JUST like realtor would do..

Actually, we got more than our monies worth because it went over schedule by 2 hours ...Ok, that part wasn't the coolest, at least not to my dog who had his legs crossed by the time we got home :P I agree with @Luc Boiron and @Larry Smet's points, and if I may add, some parts of it vaguely reminded me of 'that time in Mexico' when I got nabbed by timeshare sharks lol Sharks and such aside, it was an incredibly informative and motivating event - and yes per @Michael Ramzy, lots and tons of note taking!! 

Personally, I like to give them written notice by mail ,email and post at the property.

Search them out and send them mail.

(I closed it so to no longer accept partial payments) and that he should mail me a check to the new PO box I opened?

You can door knock or mail pre foreclosure leads  in attempt to support them in accomplishing their goals.