Sponges are very diverse group of suspension feeders, living on phytoplankton, bacteria and debris, mostly. Some species are photosynthetic, but you rarely (never?) see them for sale. I got several as hitchhikers on my live rock, and they have done well with whatever comes in with fish food, plus occasional additions of phytoplankton, like DTs, Tahitian Blend of Reed's Post-Set formula. Some have grown and spread over the years, others have simply stayed put.
As far as Moorish Idols (and almost all nudibranchs, for that matter), sponge-eaters are generally specialized for a particular species of sponge, and will choose to starve rather than eat the wrong species. You then have to face the problem that the food species is usually not known, and you can't obtain the correct species even if you knew what it was.