I can't pretend I'm an expert, but one of my first lines of attack on an integral equation is to see if I can't convert it to a differential equation and then solve using standard techniques. In your case, if I differentiate with respect to z, I get

assuming that decays to zero at least as fast as

This result is a delay differential equation. My guess is that unless you are a really good guesser, you won't be able to find a solution analytically. However, there are numerical solvers available for delay differential equations. Take a look here.

And we have now reached the limits of my knowledge on the subject.