Is there a way to search how many litters a stud dog has had?

While not perfect, you could look at OFA at Offspring and count the number of different "series" in the AKC registration numbers. Though this process wouldn't work for new studs with litters where the pups aren't of age for OFA's yet.

How about calling the stud owner?

For a fee you can order a progeny report on the AKC website. You can find it in their online store.

The AKC progeny report only shows those get that have earned an AKC title, not all get produced by that dog.