I believe there are two decision points that determine whether you receive your payments via paper check or electronically: the bank or vendor that supplies the bill pay system, and the individual bill payer. In my experience, it's more often the bank or vendor that you should try to contact.
Some bill pay systems simply don't offer anything other than check payments. Convincing them to change their payment mode could be a challenge. Others may offer both check and ACH payments, but are less aggressive in seeking out electronic recipients. If you can identify these payment system providers by examining the checks you're receiving, you may be able to contact them to dialog about getting electronic payments set up.