How to connect a switch to a wireless network?

Use a Wifi AP. Most can be run in client mode, and connect to the modem's AP.

I've used an AP set up in bridging mode to connect a switch to a WiFi environment, it worked just fine.

linksys routers will do that (that is all I have experience with). I currently run a Linksys WRT54GL v1.1 with Tomato firmware. This is an upgrade from the stock linksys firmware. It unlocks a few more features than the stock firmware will. If you want an even more advanced firware use dd-wrt. alot more features, alot more complicated. I have used both and for my needs, tomato works well at home.

DDWRT running on a Linksys WRT54G can act as a repeater. I have never attempted it before, but have a look at the documentation. I would think it would work.

Its at dd-wrt.com