Yes. From the beginning of the introduction of solid foods, a baby can eat the whole egg, even if the baby is considered at high risk for allergy.
For many years, it was recommended that babies not be given egg whites before one year of age, as it was believed this would protect them from egg allergies. However, there is absolutely no evidence that this is, in fact, true.
On the contrary, some studies show that the later an allergenic food is introduced into the diet, the more at risk is the baby of developing an allergic reaction to that food.
This recommendation applies for all babies, whether or not members of their immediate family have allergies.
Note: It is important to cook the egg completely before feeding it to the baby and not to give them runny yolks.