You should definitely let your family members know that your son is severely allergic to eggs, and itís perfectly acceptableóand even advisableóto ask them to refrain from bringing eggs into the house while youíre on vacation. I donít think itís a huge imposition for anyone to eliminate eggs from his diet for a week or so, especially when it could put your sonís health in danger.
It might be wise to call or email the members of your family before you all leave for vacation to remind them of your sonís allergy, and to ask if they would mind helping you keep your son safe. This way, if they happen to stop and pick up groceries before they arrive at the vacation house, you wonít be faced with the unpleasant task of confiscating their eggs upon arrival!
I think most folks these days are very familiar with the problem of food allergies, and with the danger they pose to sensitive individuals, so it probably wonít be difficult for them to understand your situation. Youíre obviously a very kind and sensitive person to consider their feelings, and I have no doubt that you will ask your family members in such a way that they will be delighted to help you keep your son safe and healthy while youíre all on vacation.