If you care about privacy why not make a wallet with Desktop wallets like Electrum or Wasabi for lightweight wallet and Bitcoin core for a full node wallet or make a paper wallet instead.

Or you mean a Payment processor?

The best option for this is Self-hosted like BTCpayserver I'm using this as another payment gateway to accept bitcoin.

Let me know if what exactly you are looking for? Because there is no online wallet for privacy just like the above said.