The "multifunction" version of ZPP? I would like to ask if there is a name for the class of multifunctions, each of which can be computed by a probabilistic polytime Turing machine $M$ satisfying the following two conditions:


$M$ returns a correct output of the function with high probability
$M$ can either return a correct answer or "don't know", and never return an incorrect answer.


Note that although this sounds like $\mathsf{ZPP}$, it is not since $\mathsf{ZPP}$ is the class of predicates, and not multifunctions. I really appreciate any pointer to the right sources.
I think what you're looking for is Las Vegas algorithms. I do not know that the class of functions computable by polytime Las Vegas algorithms has an abbreviation beyond "Las Vegas polynomial time". 

As pointed out by Tsuyoshi Ito in the comments, $ZPP$ is exactly the class of decision problems decidable in Las Vegas polynomial time.