Buying Bissell brand products for use in a Bissell Carpet Cleaner is not only recommended by Bissell, it is the only product to be used in their cleaners according to the user's manual. In fact; using products other than Bissell brand products voids the warranty. The manuals sometimes state that it can also cause fires and electrical shock. With all that in mind, though, there are substances that can be used to perform the same functions.
Vinegar has been used for centuries as a cleaning substance. White distilled vinegar can be used on carpets. Test a small, hidden area of the carpet first by just rubbing with vinegar and letting it set for a short time. If there is no damage, vinegar can be mixed with water in the same ratio used for Bissell products and used in the cleaner.
Liquid laundry detergents can be used in place of carpet shampoo. It's always a good idea to test a small area in a place that's hidden just to be sure there won't be any problems. Once that's done successfully, though, any brand of laundry detergent should work in a steam cleaner. Mix it in a smaller ratio of water to soap than would be used for the Bissell shampoo. Try a tablespoon or two first to make sure no residue is left.
It might be surprising, but for a carpet that isn't extremely soiled, just water often does the trick. When using water, get it as hot as you can when filling the cleaner. Obviously there is no mixing ratio in the case of water. Water has several advantages. It's less expensive than nearly any other cleaner. It doesn't leave any kind of scent behind and it's perfectly safe for pets and children.
A gentle hair shampoo, like a baby shampoo can be used in place of Bissell's shampoo products. As with any kind of new soap, test a small area in a hidden part of the carpet first by cleaning it with the shampoo. If there is no color loss or other damage, then a mix of a tablespoon or two per tank of water. If that leaves no residue, more can be used on subsequent cleanings if desired.