If your wireless router has a built-in switch (has LAN ports), you could connect the beige Mac via Ethernet.
If necessary, a wireless connection can be established if a wireless Ethernet bridge (similar to Netgear WNCE2001 or Linksys WET610N) is used at the beige Mac. A wireless Ethernet bridge does not normally require special drivers, and can thus be used under almost any operating system.
File sharing between Mac OS 9 and Mac OS 10.6.x is not without complication (due to lack of support for older protocols in the modern operating system). One way could be to set up FTP (or web) servers, and then use FTP clients (or web browsers) to access files locally. Another possibility would be to carry out file transfers over the Internet, either as email attachments or through some kind of Internet drive service.
Yes, the web browser situation for Mac OS 9 is difficult. You could perhaps have a look at something like iCab or Classilla, but you may find that they will not work with many web sites/pages.