You can but I don't. You pay the crypto to Kagi through a third party pay service. Its not 100% untraceable but it's enough to keep you out of any automated tracking that Kagi is or will eventually be doing.

You can buy it anonymously, mostly through P2P exchanges and markets. But simply buying bitcoin or some other popular coin with fiat, then exchanging that to Monero and using that to buy your privacy things will work as well.

Monero is very private and anonymous.