When buying from the same onramp, which is automatically recommended, there's no need to KYC again.
The first time, you buy crypto, you’ll have to complete fiat onramp’s Know-Your-Customer process. This can be done without leaving your page! When returning and buying from the same onramp later, there’s no need to go through KYC again. The same onramp will generally be recommending, meaning you won't have to go through KYC multiple times.
Onramper does not save the information you fill in when buying crypto through an onramp. All user information is sent directly to the relevant partner that facilitates the transaction.