Theoretically, he could. Since all alchemy is based on the idea of equivalent exchange you should be able to trade something of equal value for something else. So far though, the only thing we know that is equivalent in value to a person's alchemical gate is a human life (from when Ed traded his gate for Alphonse's body and soul).
So to get his gate back he would need a couple things. First would be some way to get to the Truth so the exchange could be made. Al could provide this since he still has his gate and knowledge from his encounter with the Truth. Second, someone else would need to help perform the exchange. Since Ed can't use alchemy at all, someone else would need to perform the transmutation. While it is unknown if someone else can sacrifice something to re-create someone else's gate, I am assuming it is possible. Nothing in the show has shown that a successful transformation cannot be reversed.
The last part is the toll to be exchanged for the gate. Given Ed's character, all the struggles they went through to avoid using philosopher's stones / human transmutation and the character growth that led to him being ok with sacrificing his gate in the first place, this becomes the biggest sticking point. Ed won't do it. Nor would he allow someone to do it for him.
TL/DR: It seems to be possible but Ed would never let it happen. Life is too precious to him for him to allow it.