Your browser isn't choosing which interface your traffic goes through, the network settings in the OS are responsible for that. In this case it would probably depend on what the default gateway was set as. By default all the traffic would go through that unless there was an exception.
If this were linux the answer would simply be 'iptables' but in windows it's a tad more murky. I wish I could answer better, I'm sure someone else will have a better answer, but this should at least get you started.
FOUND IT! It is not a pretty one, but is works.
In older versions of vmware workstation an player there was a tool for configuring network adapters. It is called vmnetcfg.exe. In the newest version of vmware workstation it is still in the install package, but not installed anymore. To get it to work in my vmware player id did this:
Get the Program
I downloaded the latest version of VMware Workstation (400+MB). When it completed I opened up CMD and extracted the installer:
> cd \path\to\installer\
> vmware-workstation-versionyaddayadda.exe \e extractedfolder
Then I went to the extracted folder and opened core.cab with winrar. Somewhere in there hides a vmnetcfg.exe. I extracted it and copied it to my VMware Player install folder.
Running te program
I started the program. I had a VMnet0, VMnet1 and VMnet8 listed. I selected VMnet0 in the list, then clicked bridged and selected my wireless network adapter instead of automatic.
Then I clicked "Add Network..." and called it VMnet2. I selected VMnet2 in the list, then clicked bridged and selected my wired network card instead of automatic.
Adding the card to the VM
First I added the second virtual network adapter in the VMware player's machine settings. The first one I kept on bridged, and configured the wireless adapter as the preferred bridge adapter (my wired adapter is not clickable anymore).
I needed to add the card to the VM by hand. This meant I had to open the VMX file in Notepad or in my case another preferred editor.
In the VMX file I added two lines (if they already exist, replace them.)
ethernet1.connectionType = "custom"
ethernet1.vnet = "vmnet2"
I saved the file and started vmware player. In my network settings I see this now:
Network Adapter: Bridged (Automatic)
Network Adapter: Custom (vmnet2)
I started the Virtual machine, and it worked!
Best Answer
In general, it is the operating system, not the application, that decides which network adapter to use, based on target IP-addresses.
If you can arrange that the different adapters are on different subnets, you can set up static routes for the IP-addresses of the websites.
I guess most PCs have their IP-addresses allocated by the local router. These can be configured so that a multi-homed PC gets different subnets for different interfaces.