Dog and cat doors are a popular category for Service Central - all those pet lovers out there wanting to give their pets immediate gratification for their needs. And rightly so - we know how insistent they can be when they want to get out (or, as my cat does, wait for me to turn the bathroom tap on - apparently drinking water out of the water bowl is unacceptable to her tastes!).
The rule to the type of tradesperson you need to do this is simple - if you need to install the pet door into a glass door or window, you need a glazier. If it's a wooden door, you need a handyman.
Usually, it's best if you purchase the dog or cat door that you want installed, and then post a job for the type of tradesperson you need! However, some handymen will also be able to supply the cat or dog door if you wish.