1. 'I am sick at my stomach from eating the food' --is sometimes heard. But not, 'I am sick at eating the food.'
2. I am sick of eating the food. --fine.
3. 'I am sick from eating the food.' --okay, too.
The phrase 'eating the food' is just a bit odd. Maybe, 'I'm sick of eating fast food, or buttery food, or tasteless food, or sweets, or low-calorie foods.