My son had a severe egg allergy at that age (thankfully grew out of it by age 6) and this was before the mandatory restaurant menu labelling (early 2010s). Restaurants nowadays label (egg is marked with a C) but can't guarantee egg free. Normal products sold in grocery stores will highlight allergens in bold in the ingredient list and highlight traces too. Since egg based lecithin is more expensive than soy based you are mostly fine.
Normally if you pick foods that normally don't require eggs you should be fine in restaurants. Egg is used quite a bit in breading schnitzel for example, so there is often cross contamination if you order french fries that were fried in the same fryer. Good restaurants will listen to your request and check with the kitchen, but often come back and say they can't be 100% sure.