I doubt they're driving from the passenger seat. It's hard to reach the pedals from the passenger seat. Many mail delivery vehicles in the US are right hand drive; the driver's seat is on the right. I don't know that left hand drive is strictly required by any states, it's more a matter of there's not many models available for sale in the US with right hand drive as an option, and few people order them when available; and importing out of market RHD cars is a chore.
You can (or could) order some Jeeps in RHD, and a lot of rural USPS contractors have RHD minivans (many from the Japanese Domestic Market)
I did see a 'converted' RHD Windstar (Ford minivan) on FB Marketplace, where they removed the driver seat, put a flat platform in, and connected controls with cables and levers. That's an option too, I guess.
This is partially because the car dealer lobby convinced congress to prevent the importation of cars less than 25 years old. Ironically, they did so using safety regulations, so a brand-new car meeting current European safety regulations is legally unsafe, but a 25-year-old car, complying with what European safety regulations were 25 years ago, less any deterioration in those safety systems, is legally safe.
Most rural routes are served by the USPS-proper.