Yes. Welcome bonuses are common promotion at real-money online gambling sites in the US legal market. In fact, it's a rare casino or sportsbook that doesn't offer some type of welcome bonus.
For casinos, the typical bonus ranges from $10 to $25 and is available only for new customers. Different terms and conditions apply, so be sure to read the fine print. For sports betting, bonus bets range from $10 up to $50 and also are only for new customers. View current casino bet offers here and sports bet offers here.