Level 6
I contacted Stripe 2 weeks ago about this. They said they are working on integrating ACH. They didn't have a time table, but said they are doing a private beta right now with select users that are actually using the ACH system. Based on the information they gave me, it sounds like they should release it in the next few month's if everything goes well. They may do a public beta after their private beta is complete.
Having said that, that's in the future not today. The other solution I found a few weeks ago, but haven't had time to look into too much detail on is Forte ( https://www.forte.net/developers ). I know they have a RESTful API. I haven't looked to see if they have PHP Libraries ready to go.