i bought one of those hand held carped steamers, cuz we rent too, and our carpets were brand new when we moved in....i have found that just water does an amazing job. for whatever soap you end up using, you COULD just put a tiny diluted amount on the grease stain only, and let is soak, then suck it up. no need to put soap in the steam vac, IMO....you can just use water. i use natural laundry stain removers. test a small hidden patch of carpet fisrt, but i used "Bio Pac laundry stain remover" on my carpet (i diluted it) and it worked really well. however, it DID seem to look lighter where i used it, but i am wondering if that is cuz my whole carpet is sorta dirty looking, so its not lightened, just cleaner. it was pretty subtle, and now i cant see the lighter spots at all..if you are renting a big steam vac and doign the whole carpet, this might not matter at all , since it will ALL look lighter when you are done...and be more even because of that.
for what its worth, .i am trying BAC OUT next time, it says its OK on carpets. (simply because i just bought some for other cleaning needs)