Ride-share platform Uber and an SEIU local asked Washington state leaders to help create portable health-care and other benefits for gig workers, who often lack them because they’re not considered employees.

“At a basic level, everyone should have the ability to protect themselves and their loved ones when they’re injured at work, get sick, or when it’s time to retire,” the platform and union said In an open letter to state legislators, unions, and businesses, posted on both Uber’s website and the local’s.

The letter was signed by David Rolf, president of Service Employees International Union 775 in Seattle, Uber...