Let me first clarify to make sure I know what you're looking for:
- you want to be able to email the entire group by emailing to a single group email address
- you want to dynamically update the group with new users from your web backend
- you want to do all of this without running your own mailserver
... I'm not sure you can get all of that.
What you're looking for is a 3rd party mail host that provides you with an API that allows you to dynamically update group membership. Google Apps may allow that, you'll have to check through their API references (a promising place to start), but I'm not sure if that's available on the free tier or not. I do know that you can include non-domain email addresses in the groups (i.e. if you sign up for Google Apps under example.com, you can send to email addresses that aren't @example.com). But, after a quick perusal, I'm not sure if they allow quite that level of access, and it's going to be some slogging through to even figure out if you can do what you want.
If Google Apps doesn't do it for you, here's a (possibly dated) list of email services (most likely non-free) that provide an API: http://blog.programmableweb.com/2007/12/05/16-online-email-apis/
If you determine that one of the 3 points above is not important, then the job becomes easier. If it's OK to email each person individually, then simply manage their group memberships as suggested by other answers here, and loop through their email addresses to message them. Likewise running your own mailserver.
Why is it (if I've understood correctly) that it's important to email just a single group address?