Depending on the direction you're coming from (I come from the West) the town of Indio is ~21 miles West of the off-ramp. There's a TA Travel Center, Love's, etc., right off Interstate 10 that has propane available. The town of Blyth is ~72 miles to the East right off Interstate 10. Unfortunately, I haven't been there, but a quick Google search indicates there is propane available.

Chiriaco Summit gas station (Chevron) up the I-10 from this campground has propane.

There are several options in the town of Joshua Tree on the north side of the Park. Tractor Supply Co. is my "go-to"... but check others for best price.