It would annoy me on many levels, and there are several variables.
If the kids are too young, then there should be none of this at all. I wouldn't want my kids making me the 'bad guy" for saying no to anything - whether it's church or Chuck E. Cheese or a party or the movies. If we have something else on the schedule, I want to decide on whether it's appropriate, fun or convenient. Or all 3.
If the kids are too young to be relaying accurate invitations at all, then it's inappropriate due to their age.
Religion adds another layer. If there is an evangelical component to this (as someone mentioned below, rope in the kids and the parents will follow, or "let's convert the undecided"), then there's an entirely new aspect to the inappropriateness. It's manipulative and deceptive.
I think I'd start on the lowest level - tell the mom you'd appreciate it if all invitations came through you since there are potential conflicts in both scheduling and interest level. You don't want to be the bad guy if it's not something you can let them participate in, and you don't want them to be put on the spot every time about saying no to something that doesn't interest them. Tell her you appreciate her welcoming and inclusive attitude, but you'd like her to respect your role in determining their activities and schedules. I don't think you have to open it up to a big discussion about religion unless and until she says that's her overriding purpose. I'd handle it as a "protocol" and "parenting" issue first, of having adults handle adult things.