In my opinion, the simplest way would be to have a virtual machine (i.e. Virtualbox/VMWare/VirtualPC with Windows) which you would run your OSBot + OSRS client on and then configure the virtual machine to use the VPN. It adds some overhead and is a little bit of hassle to set up but it should work just fine. At least you would be able to control the bot without walking to your other PC.
Without running a virtual machine, I don't really know how you would configure the VPN just for a single application. It should be possible with the proper firewall settings but I have never done it (relevant answers: http://superuser.com/questions/294008/use-vpn-connection-only-for-selected-application). Another way would be to try and see if the OSRS client will accept proxy settings through command line options but I have personally never tried it either (some relevant answers: http://stackoverflow.com/questions/120797/how-do-i-set-the-proxy-to-be-used-by-the-jvm). Note that this method would need a normal (SOCKS) proxy, not VPN.
EDIT: You can also try what Slawa said. I'm not sure if it's enough to just configure the proxy for the browser, though. The browser will use the proxy for sure but I'm not sure if the Java applications will. So if you're not careful, you could leak your real IP out and the proxy would be of no use. But I don't really know, maybe Slawa has tested it out. If it works like it should, it's probably the simplest way.