Dominion is right, for this, don't use a harness for a cat's inspection. Just get a new carrier. The vet will be thankful. It's also important to think about dogs and other animals, that could freak your cat out, making her frightened and scared, or aggressive. This is something to avoid.
For normal harnesses, say if you want to walk her, which is fine after she gets her shots and everything [!!!], harnesses are weird but usually simple once you figure it out.
There's usually a metal loop on top, meant for hooking the leash onto, so that's where you start. It somewhat varies, harness to harness, but there's a solid collar meant for going around the neck, a smaller piece without the metal loop that goes under their ribs, to hold the two parts that clasp together, usually in a buckle kind of seatbelt arrangement kind of lock to hold it all in place. Make sure it's tight enough that it doesn't move but you can comfortably fit two fingers in-between all the areas that touch the cat so they can breathe.
Do this while the cat is calm. If you need more help than that strange description I gave you, use the specific instructions the harness has on the packaging [if it doesn't have anything, don't buy it], and ask the pet store workers for help, and ask your vet too while you get her vaccinated.
But please don't go in there with her on a leash, that's asking for SO MUCH TROUBLE FOR YOU AND YOUR CAT.
