Yes you can use Simple Green cleaner or any other cleaners in your carpet cleaner to clean the carpet and remove stains from your carpets.
Just fill the carpet cleaner solution tank with with half Simple Green cleaner and half water and then use the carpet cleaner as normal.
You can also add vinegar to your carpet cleaner solution tank to disinfect and clean your carpet as well.
If you have a really stubborn stain on the carpet you can spray or pour some Simple Green cleaner directly onto the stain and then use a scrub brush to scrub the stain out after letting the Simple Green cleaner sit for a few minutes.
Then go over the stain again with the carpet cleaner with the simple green cleaner in it and the stain should come out of the carpet.