I don't see that as likely. First, there's not really much real world precedent for that. If Lady Gaga were to die, I wouldn't expect her best friend to be like, "I will become the new Lady Gaga," and I don't think that would go over too well. Great athletes are honored by having their numbers retired, not by having other great athletes wear them. Second, as a former assassin, Bucky isn't exactly the most qualified to take up Steve's mantle, so I think he'd be uncomfortable with doing so, and I think the others would be uncomfortable with it, too.
Now, Marvel could make it happen if they wanted, but in that case they'd be doing it because they wanted it, not because it would be the most natural outcome.