Internet help.

[Oldsmobile_Mike]

Yes!  You can do this with DD-WRT firmware.  Grab yourself a cheap old Linksys WRT54G router (probably cost you next-to-nothing on your local craigslist), flash it with the firmware update, program it to "receive" through the antennas instead of "send", and you should be all set.  I've used this successfully in many locations for years and it should be plenty fast enough for an Amiga.

Here's a brief guide but there's many other sites out there with more complete instructions:

https://www.dd-wrt.com/wiki/index.php/Client_Mode_Wireless

I'm sure other routers can be configured in this manner, I'm just  recommending this particular solution because I've used it successfully in the past and know  it works.  

Now queue someone recommending Powerline Ethernet in 3... 2...  :p

[Oldsmobile_Mike]

Hummm...  24/7 BBS?  You might want to just run cable for that, depending on how much bandwidth your users consume and how much latency they'll tolerate.

Maybe you can snake a wire up through a heat vent, or drill a small, inconspicuous hole.  If it were me doing it I'd just run all the cables before the drywall goes up, but I'd imagine you'd have to negotiate with your contractor for that - and they probably won't let you, I suspect.  Those guys can be kind of fussy depending on your relationship with them.

I'd still avoid Powerline Ethernet like the plague.  

[Oldsmobile_Mike]

I flashed the DD-WRT firmware last night and everything seems to be working fine with the routers about ten feet apart. Will have to wait and see how it works at the new house when they are further apart. Thanks again for the suggestions!

Fantastic! One other thing you can try if you want to squeeze out a little more range is bumping up the transmit power in DD-WRT and using bigger antennas. Back in the day I had a few that were 3' long, stuck up on a pole to run WiFi out to a facility out-building (about 1/4 mile, line of site). Still use a similar method to blanket large areas (I.e., warehouses).  Good luck!