The answer to this question is: YES.
It’s not necessary to have a Social Security Number (SSN) to obtain a credit card in the U.S.
Can I apply to all types of credit cards without a Social Security Number (SSN)?
No, not all U.S. financial institutions (banks, financial cooperatives, among others) have credit cards for people with no Social Security Number (SSN), but there are a few institutions that do offer this option and these can be reviewed and compared.
What information do I need to provide if I don’t have a Social Security Number?
This information can vary per financial institution, but generally speaking, they all ask you to confirm your identity with a government official identification (like a passport copy), U.S. address and income proof. The rest of the solicited information varies per application.