Can we use OpenStreetMaps to replace Google maps (apps and web)?

There is not a "single" OSM API that you can use (matter of fact what to usually is referred to as the OSM API is actually for editing only), but a number of different solutions from which you can pick.

I think a good starting point would be the developer section of the OSM wiki https://wiki.openstreetmap.org/wiki/Develop in particular https://wiki.openstreetmap.org/wiki/Frameworks

It is possible after first checking for existing address nodes and possibly moving it to the correct location. If the client pin location is within an existing address polygon, there should be no action. If the address is already defined for a building at a different location, it may be better to create a note: either the original building address has been mis-marked or there can be multiple buildings with the same address.

In any case it is important that the email for the OSM account used to set the address node be monitored for feedback. It is likely that the clients confirming their address location won't have detailed OSM knowledge to be able to respond to questions about their edits, so the best solution would likely be a single OSM account administered by you.