Well the good news, Safari is most likely not the problem. The bad news, I think there's some confusion on what version of what you're running. So let's go over some terminology:
1.) If you're running OS X 10.6.x then that's Snow Leopard, if you're running 10.7.x that's Lion and if you're running 10.8.x that's Mt. Lion.
2.) If you're running Safari 6.x.x then that machine is on at least 10.7.x (Lion). Safari 6.0.2 is the latest and greatest version so that's up-to-date. Is the laptop running Lion or Mt. Lion?
3.) The iMac is running either Snow Leopard or Lion as Mt. Lion can't run Safari 5.1.7 but the OS is irrelevant for the purposes of troubleshooting as 5.1.7 is the latest version.
There's no mention of troubleshooting in Facebook video chat's documentation other than the standard and customary. I downloaded it on a 10.6.8 machine running 5.1.7 and it fired right up immediately. I do know that it runs on Java and there was a recent-ish Java update. So that'd be the next thing to check.
You can try downloading it and installing. - http://support.apple.com/kb/DL1572