I don't expect people to provide special treats for my GF child at their parties, and you shouldn't be expected to make a cake without eggs. His own mother ought to send something with him, just like I always bring something special for MY child. You've illustrated why this is very easily...you are unsure if someone who is allergic to eggs can eat them if they are cooked.
The answer would be no, first...if that were the case, we wouldn't have a problem, because I don't know too many children eating raw eggs. Not trying to be snotty, either, but that's just it...before I had a child with special food requirements, I was totally clueless too because I really didn't NEED to know. ;)
You can find egg substitutes in health food stores or in the health section at large chain retailers, but as Rosecity mentioned it's very expensive...and you'll use very little.
The soda idea is interesting...if it works, then great!
But I would still call his mom and ask, because perhaps she intends to send him something. I always do...yup, I show up at people's weddings carrying a piece of my own cake for my kiddo. ;)