There's several ways you might set up a system for a team of people to be able to add an adjust "boundaries".
It depends how much data there is but I think you should probably tackle this with polylines overlaid upon a basemap. You might run into problems with this approach if you have a very large number of boundaries or the boundary polygons are very detailed, with a very large number of corners. A web browser can start to get a bit overloaded if you present too much map data as a vector overlay, but you can go quite a long way with this kind of approach.
There's a nice tool called uMap which lets you create a map with polylines, in a very easy point and click way, and then open this to a group of people to make edits.
That should give you some idea of what's possible in terms of editable polylines, but I don't think it will give you the naunced permissions model you want, or the feature "identify if an address is within a defined territory". It is open source though, so could be customised or...
develop something like this yourself from scratch using a web mapping API You're asking about APIs so I assume this is not a crazy suggestion. My personal recommendation for this would be to start with leafletJS and OpenStreetMap. Very basic example of drawing a polyline.
OpenStreetMap generally doesn't attach any conditions to data overlaid upon it, unless you are copying ('deriving') a substantial chunk of geodata by tracing from the background map, in which case it must be ODbL licensed. Probably not in your case. Google by contrast have some fairly aggressive terms of use on their API, granting use of your data to them. Most web developers are blissfully unaware of this. And google does not permit any copying of their maps whatsoever (this is the big "open" difference with OpenStreetMap actually)
In both cases, when it comes to overlaying polylines, you own the data you're overlaying, storing it on your own server, and commercial use of the base map/APIs is allowed.
When it comes to building it yourself, the "identify if an address is within a defined territory" feature gets a bit more tricky, but it's also the kind of feature you could present with a nice UX if you build it yourself. Otherwise a tool like ArcGIS online might allow your users to do this kind of calculation but would also confuse the hell out of your users, letting them do all kinds of other calculations. So to build it yourself, have a look for "geocoding" APIs to go from address to lat/lon. Then maybe have a look at turfjs for calculating (client side) which polyline area a point lands within. That's just one possible approach
Hope that helps
