I was thinking about making a civ with the grazer tag, and came to ask questions, didn't find answers by searching, and realized I might as well test it myself.
As it turns out, grazer for player civs works out... okayish.
I gave dwarves [grazer:1000] and set FPS to unlimited. Embarked with no food. They all ran out of booze then dehydrated to death.
I gave dwarves [grazer:1000] and [NO_DRINK] (don't think they effect each other), embarked with no food, and they survived years and years on the grass alone.
I gave dwarves [grazer:1000] and [NO_DRINK], did not bring food, dug hole and made it a meeting hall. Dwarves came in, and eventually got hungry until one starved to death. I then made another meeting hall out on the grass, and they all went there and stopped being hungry.
From the three tests, it appears that grazer dwarves do eat grass, can subsist off of pure grass, but have no instinct to go seek out new grass.
My understanding of the grazer tag is that it does not affect hunger, just changes how much the hunger counter is reduced by eating one grass. The wiki says that it is 100,000 hunger ticks, but that would mean my starved dwarves would have to eat 100 grass each to recover from near-starvation. It did not appear anywhere near that.
One assumes underground grass/moss/fungus should work fine. Still need to test if they will eat food normally. Still need to test what organs, if any, are necessary for grazing. (Is a mouth or something needed for eating in general?) Etc.