Yes and no. If their computer has a firewall (and you don't have access to their computer), you cannot tell what programs they are running. That's one of the purposes of firewalls.
However, once they leave their computer, you can detect this.
One way is to put a sniffer on your network. That will tell you what traffic is passing by. Although there are plenty of free sniffers out there, very few will put your network card in promiscuous mode (the mode it needs to be in to see their traffic).
Second, try checking your router. Many keep logs of all the connections they make.
Third, install a proxy server. That way, all web traffic would have to hit the proxy in order to get out to the Internet. That way, you can see everything.
Finally, you could install a DNS server on your network and point everyone to it using whatever DHCP server you use. All requests for web traffic would then be routed through your DNS server which would cache the responses. I use one to speed up web browsing (my ISPs DNS servers are not that great), but you could use it to check traffic through the cache if you wanted to.
FYI: Most businesses use option 3.
Courtney