Among the 560 owner complaints naming this part, rim is named in 7% of them, more than any other part.
Other parts named in the same complaints. Rim: 7%; Tire installation: 4%.
Owners name the rim most often in the same complaint, which is not proof that one caused the other. Ask the shop whether it checked the rim too, so you are not quoted twice for one visit.
Data and chart by CalcScrape: Tire Pressure Monitoring System Relearn Cost Calculator