Yes mixing vinegar and water together in a Bissell or any other carpet cleaner is just fine and Vinegar makes a very good natural cleaning agent for carpets, upholstery and much more.
If the stain is really bad then you can spray vinegar and water onto the stain and let it sit for 15 to 30 minutes and then use vinegar and water again in the Bissell Carpet cleaner to clean the carpet.