I think this really depends on the restaurant (and maybe the time of year you go). For example, when we were in AK, it was about 11:00, we weren't hungry but I was feeling really sick and wanted to get outta the heat and into some A/C and maybe get some water. We were in DinoLand so we headed to Restaurantatorus (or however you spell it) and they were closed until 11:30. We checked the Times Guide and it said that Tusker House started serving lunch at 11:00.
When I've been during peak season, I've found there was always at least one counter service place serving lunch by 10:30am. It may have even been 10am. When you first walk into the park, grab a times guide. It should tell you what time each counter service starts serving lunch.
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