Fingers crossed that you get those permits!
Going up Half Dome via the cable route is something that I’ve done twice. The only piece of gear that I used were leather gloves (nothing fancy). I did see some people with Via Ferrata lanyard, but I didn’t notice if they were wearing a climbing harness. What I’ve heard is that nobody rents these out due to liability issues. You may be able to find a harness and Via Ferrata lanyard used on Craigslist or FB marketplace, but you’ll need to know how to properly wear the harness and use the lanyard.
Know that the vast majority of people to do the cable route up and down Half Dome use no clips or harness and have zero problems. Using clips can actually complicate things, distracting you from the surroundings and introducing risk.
There’s a great blog post you can read on this very subject here. https://yosemitehouse.com/half-dome-safety-gear
My most recent trip of Half Dome I was accompanied by my girlfriend (now my wife). She got part way up the cables and panicked. I was up ahead of her, turned around and came back down to where she was and talked her through it.
The key is to don’t worry about being slow or holding other people up. Wear leather gloves (the steel cables will have “frays” that you don’t want to get into your hands). Work your way from one 2×4 to the next, stopping and resting at each as needed. Take your time.
Going back down, use your gloved hands as brakes to control your speed.
And most importantly, be aware of conditions. Don’t attempt it if it is raining or even wet. The most important piece of safety gear is the gray matter between your ears.
I’ve attached a photo of my wife climbing up Half Dome (she’s wearing the blue tank top and facing the camera). As you can see, there are 2x4s where you can stand and give your arms a bit of a rest. She made it to the top!