Multiple protests have been held at vaccination sites this year, including Dodger Stadium.
Senate Bill 742 would not outlaw protests themselves, but would make it a crime to bother or interfere with people receiving their shots at public sites.
In the bill's current form, violators could be subject to a $1,000 fine and/or six months in jail.
Some legal experts have questioned whether the bill would violate First Amendment rights.
The U.S. Centers for Disease Control and Prevention has maintained the COVID-19 vaccines are safe and effective. The CDC also has said that people who have already recovered from COVID-19 should still get vaccinated.
The Associated Press contributed to this report.