There will be ski shops all over Kitzbuehel to rent equipment.
Prices at most European ski resorts are not heavily discounted, but pretty inexpensive to begin with. Looks like 149 EUR for 3 consecutive days during your time there, which is less than 50 EUR per day, or just about $60/day, Compare that to Vail.
https://www.kitzski.at/en/tickets-prices/tariffs-winter/2-15-day-ski-pass.html
If they offer some kind of rescue insurance, take it. It is usually only a couple of EUR per day. Ski Patrol service is not free in Europe.
Great skiing, by the way. I have not looked at snow reports. You can ski the Hahnenkamm race course minus the Mausefalle (it is roped off, it's a cliff) to appreciate what the racers do. If you can ski it without stopping, I'm impressed.