Until the 1970s, full service was the norm, with self-serve gas stations being a rarity. These days, it's the other way around, and full service is very unusual but most common in wealthier neighborhoods, where the privilege of having someone pump your gas for you will cost you a per-gallon premium on the gas, plus a tip to the attendant. True full service not only includes filling up your tank for you with the appropriate grade of gas, but often the pump jockey will also wipe your windshield and check your oil and tire pressure.
The Americans With Disabilities Act requires self-serve gas stations to provide equal access to their customers with disabilities, which if necessary means providing refueling assistance upon the request of an individual with a disability (unless the station's operating on a remote-control basis, with a single employee), but for able-bodied drivers, these days it's pretty much a DIY affair.
Until you posed this question, we weren't aware of any full-service gas stations anywhere in Las Vegas, but some googling did pop up the Boulder Basic Sinclair station (432 S Boulder Hwy, Henderson), which basically offers what's known as "minimum," as opposed to "full" service -- they told us that if you go inside, or make yourself known to a member of staff outside, they'll pump your gas for you at no extra charge. They're not going about to go to the length of your oil or tires, however, nor bring you out any snacks from the convenience store.
This is nowhere near the Strip or the airport, either, and while the photo below, taken in 1968, reveals that the heart of Las Vegas Boulevard was home to more than one gas station back in the day, today it's a very different story. However, we placed a call to the Circle K station located at 2308 South Las Vegas Boulevard (i.e., the Strip, at Sahara), and the very helpful member of staff we spoke with said they don't offer full service as the norm, but if you asked them to pump your gas for you, they'd be more than happy to oblige.
Whether we just happened to luck out and get the friendliest member of staff is another matter, however, and we can't vouch for everyone there being so accommodating, but it's worth a shot if you don't want to take the edge off your relaxing vacation by having to pump your own gas (heaven forfend!) on the way out of town. The Chevron at 3201 W. Tropicana (just off the Strip) was equally obliging, whereas the Smart Mart at 4207 South Las Vegas Boulevard was pretty adamant that the customer pumps their own gas, as was the Terrible Herbst at the Strip and Sunset.