Yes, you can bet on sports without a social security number. Social security numbers are rarely needed for in-person sports betting, which is different than withdrawing money.
You can walk into a sportsbook and make bets without your social security number, but if you plan on withdrawing more than $600, you'll need a social security number for tax purposes.
The main piece of identification needed for betting at sportsbooks is a driver's license because four states require bettors to be at least 18 years old and the rest set the limit at 21. Even where 18 is the limit, there are some casinos that abide by different rules and are 21 and over no matter the age limit for sports betting.
It's a slightly different situation online because operators like DraftKings and FanDuel require the last four digits of your social security number to deposit and sign up for an account. The main reason it's needed is because if you win over $600 at an online sportsbook, they are required to report your winnings.