I help run a drop in style church play group. We are not specifically for vulnerable families, but we do attract a lot.
If you are setting up a group specifically for vulnerable families, have a good think about how to make your group accessible to them. What are the barriers that will stop them walking through the door? Is your venue somewhere they can get to easily (walking or public transport) or is it hard unless you can drive? When they arrive, is it obvious where they should go and what they should do? If they are late, will they still be welcome?
I'm of the opinion that the welcome people get when they walk through the door is one of the biggest factors in running a successful group of this nature. It's a scary thing to walk into a new group where you may not know anyone and are unsure about what to do. So make sure you have someone on the door, greeting people with a smile and a welcome, showing them where they can park their buggy and where to sign in. Make it easy for people!
For a group for vulnerable families to be successful, I think you also need to be mindful of not accidentally creating a "them and us" attitude (- an attitude that you're doing something for them rather than something with them). This is quite a subtle thing, but people respond best when they're not made to feel like charity cases. So talk to people, get to know them and what's going on in their lives. Try to leave your prejudices aside when you internally wince at some things. Don't foster an atmosphere where people feel judged if their child won't sit nicely for the singing or whatever. Making the group free is not necessarily a great idea either, however much you might just want to help people. A small nominal charge per family can help people feel like they have some investment and ownership in the group.
I won't give lots of ideas about set up and activities, because lots of others will do that, but I will say that by far and away our most popular activities are the play dough table and the sand tray!
All the best Fluffynotebook - it sounds like a very exciting opportunity!