So you want to use Reaper (a DAW: Digital Audio Workstation) to mix the band, and feed its output to OBS as an unchanged passthrough? For that, you need a loopback audio device. Lots of those available, if Reaper doesn't have one built-in.
A loopback is just a virtual speaker and a virtual microphone, all in software. Whatever goes to that "speaker", shows up in that "mic". Simple as that. Connect Reaper to the loopback speaker, and OBS to the corresponding mic, and now you can feed Reaper into OBS.
Lots of people seem to like VoiceMeeter and its parent company, but it's certainly not the only one:
VB-Audio Virtual Cable and App's
vb-audio.com
That's a single loopback, which should be all you need if I understand you correctly. Or you can look at the other tabs, which add more loopbacks to the same system as well as a little bit of processing along the way. But since you're using a DAW already, it'd be better to keep all of your processing in there.
That said, there might be an even simpler way:
Send Reaper to a physical output, possibly even put headphones there, then use OBS's desktop audio selection to listen to that output. That's effectively a free loopback that the operating system gives you for every output device.