iMac "lampshade" USB1.1 or Ethernet OSX 10.4 Tiger wifi adapter

If you have one of the AirPort Express 802.11n models, it may be possible to use the device to join an existing Wi-Fi network (acting as a wireless Ethernet bridge). An AirPort Express needs the appropriate Apple software for the setup (whereas a third-party Ethernet bridge typically makes use of a web-based interface). Regarding WPA/WPA2 security and other questions in connection with an AirPort Express, I recommend the AirPort area.

An 802.11n (but not the g-version) AirPort Express should normally be able to operate as a wireless Ethernet bridge (that is, as a wireless client). If so, and if necessary, the setup could be done on another computer having the appropriate Apple utility. The guys in the AirPort area are the real experts regarding all things AirPort Express. About WPA or WPA2 security, see the replies to a question I had earlier: WPA2 possible when AirPort Express is used as a client?.

Otherwise, you could of course try an inexpensive device like the Edimax EW-7811Un 150Mbps WiFi USB Wireless Adapter with the appropriate drivers (http://www.edimax.com/edimax/download/download/data/edimax/global/download/for_h ome/wireless_adapters/wireless_adapters_n150/ew-7811un) to begin with, as an experiment.

A third-party wireless Ethernet bridge (such as one of the models mentioned above) would be a good universal alternative (without the need for special drivers or setup software), but will also be more expensive than the USB adapter.

Using a USB Wi-Fi adapter is possible, but you will notice that only a few modern devices (with WPA2 security) support Mac OS X 10.4. USB adapters depend on special software, so make absolutely sure that the supplier can provide you with the required driver software (or at least a valid download link). Mentioning a certain adapter model here could be misleading, since a hardware revision may or may not work with previously available software (but Edimax EW-7811Un at least appears to have had drivers for 10.4). A USB 1.1 port will limit the speed (to less than 12 Mbps).

A wireless Ethernet bridge on the other hand does not (normally) need special driver software, and can thus be used under almost any operating system. WPA2 security is standard on modern devices. You will find a nmber of wireless Ethernet bridge models that should be suitable in this case. You may want to have a look at, for example, Linksys WET610N and Netgear WNCE2001. Even some Wi-Fi range extenders with an Ethernet port (such as TP-LInk RE200) have wireless bridge capabilities.

Addendum

The information about ProxySTA (under Express exclusives) in an older Macworld article (https://www.macworld.com/article/1132754/networking/express11n.html) could perhaps be of interest to you.