You can of course test the unit's mixer functionality without a computer. In fact I have used one of my Tascams, the FW 1082 as a dedicated monitor mixer for several sources. But that's not the specific answer you're looking for.
Surely you can obtain a firewire card, or a computer with firewire, or someone you know who has one? It may be best to be on Win 7 so you don't have more trouble than need be getting it running.
Another option, is you can buy a used computer faily cheap on ebay etc, that's what I did. I bougt a used Lenovo ThinkStation with Win 7 Pro, it works great with my Tascam FW units.