They have soup & salad as appetizers. IIRC the choices were vegetarian chili, tossed salad or a soup of the day. The chili is good but not for picky eaters - it has a lot of veggies in it, including some that you wouldn't normally find in chili, like olives. I really enjoyed it but my slightly picky DD thought it was "weird".
At the Plaza, the sandwiches are more than big enough for sharing, so your 2yo should be fine with sharing from your entree even if there isn't an appetizer that would appeal to him. DH & I agreed that if we ever do Disney without a dining plan, the Plaza would be one of our must-dos because we could easily feed our entire family (11yo, 7yo, 1yo) with 2 adult meals and 1 banana split!