I'm moving to a new apartment and currently have no internet service. My neighbors were generous enough to allow me to connect to their wifi AP. I want to use a RB815G-2HnD to connect to the wireless AP and route all internet traffic from the LAN ports through it.

Is it possible to do this?

Yes it's possible, simply make same settings as neighbor AP, in wireless security profiles, and wireless interface. change mode to station.

Add DHCP client on your wireless interface.

Then configure DHCP server for your LAN interfaces.

If you want your own internal LAN, make sure you put DHCP on your LAN bridge or master port and make sure the WLAN interface is not member of the bridge.
Also make sure you create a masquerade rule for the traffic on out-interface=WLAN.

If you do not really need your own dedicated LAN IP segment, just put all interfaces into the same bridge, including the WLAN interface and let your neigbours AP deliver an IP towards your devices.