but you could try making a paste with baking soda, scrubbing that into the spots (the whole thing?), and then with a stiff brush scubbing vinegar in. Make sure to wear some form of gloves (I'd recommend the heavy latex gloves for scrubbing toilets.).
The two will break down, so don't mix them together before using. (as usual mixtures online... they tend to break down into a less-acidic version of vinegar.) as used here, the idea isn't to make a super-cleanser from two already-good cleansers- you use one to scrub in and get to stuff, and then use the other. The foaming action from the reaction helps lift greases and stains out of the fibers, though.
alternatively, just scrub with hydrogen peroxide, which as a heavy oxidizer is great at attacking organic stains.