This appears to be convex, but I don’t see how ti implement it in CVX. Your last hope probably is that one of the Mosek guys can figure out how to do it, and if not, it’s a candidate for @Erling’s challenge.
Note that the related function, y*exp(y) is not convex for y < -2, and therefore is not convex over its “natural” domain. I do not know any way of implementing that in CVX even for y restricted to be >= -2 (or 0).
The function abs(y)*exp(abs(y)/x), although apparently convex for all y and x, would appear to be no easier, and perhaps more difficult, to implement in CVX than y*exp(y) for y >= -2.