I walk or bicycle, Wife used a mobility scooter (Way smaller than a golf car) this is similar to the scooters offered by many stores for customer use.
Depending on the size of the pick up cover you might slip a cart under it, else, a bigger cover or remove it.
Side note; Since we full time wife used Campground laundry
I had a lawn & Garden wagon (this is a little green wagon with removable sides, kind of like a little red wagon save the sides and bottom are mesh)
Modified it to be towed by the scooter for taking laundry to the machines and returning with it,, among other things.. worked well.