Is there any service or delivery job in the US where officer style caps are worn?

Caps like these that used to be the hallmark of service personnel. Gas stations, milk delivery men, ice delivery, would wear these caps, but you never see them worn by service personnel anymore.

Other than military or law enforcement and related fields does any service profession use caps like this these days?

Doormen?

In Chicago, Metra train conductors wear these hats: http://www.examiner.com/images/blog/EXID23824/images/resized_DSCF0195Metra.jpg

The Maytag Repairman?

Amtrak employees often wear similar caps.

Doormen.

Limo drivers. Probably some bus drivers.

NYC subway conductors. Lot’s of other train conductors.

Airport porters. Train porters too.