We got several packages a month mail-order. When I was a kid in the 60's we lived on a farm in central Illinois. My parents and grandparents did a lot of catalog shopping. USPS used to deliver packages frequently. Big mail-order businesses at the time included Sears, Fingerhut (first catalog in 1948), Hammacher Schlemmer (first catalog in the 1800s), JC Penney, Montgomery Ward, Spirgel, and more. Most of these places had accounts, but you had several payment options.
Most common method of ordering was to pull the order form from the catalog, fill in the items you wanted, calculate the costs yourself, and send it with a check. In those cases, often orders shipped once your check cleared the banks. In other cases, you could order and be billed later. Sometimes things came COD. Sometimes they came with a bill inside the package. Or a bill would arrive separately from the package. We did the same for ordering parts for some of the equipment, which wasn't available locally.
Most mail-order companies had customer credit accounts, and you would just list your account number on the order form, or it might be pre-printed because the catalog was shipped directly to you. Some, like Fingerhut, used to put a peel-off mailing label on your catalog. It had account information printed right on it. You just pull it off the waxed backing, and stuck it right on the order form, which was inside the catalog. They would ship the order to you immediately, and you'd get a monthly bill. Some, like Penneys and Sears, offered their own credit cards, and you would just use their cards to order. A lot accepted Bankamericard (which became Visa).
Every adult I knew had at least one credit card in the 60s. I think the most common were probably the gas company cards. Shell, Fina, Gulf, and the like, although there were plenty of bank cards floating around.
Our normal mail carrier was a nice lady, and she drove her own car. Most days she drove a station wagon because she had so many packages to deliver to homes and farms along her route, but some days she drove a little car if she didn't have much to carry. I remember in the '70s when she got a new Jeep Cherokee and was so proud of it. The first day she drove it, she stopped to talk to Dad, and the back end was almost full of boxes for delivery. Around the holidays she would sometimes have to split her route up into thirds because of all the pre-holiday catalog shopping, and she would sometimes drive a full sized van.
UPS didn't deliver out where we were at the time, too far out in the country until the 70's. It was "too far off their regular route." We were 3 miles from a small town, and 10 miles outside the nearest city. Go figure. If something came by UPS, my parents would have to drive into town to pick it up. I remember those rides quite well. Dad commented once that they probably couldn't afford to deliver outside town because they charged less than USPS did for delivery, and yet the companies that shipped by them didn't know that and therefore used them a lot. Mom was always making notes on the order forms to please ship by USPS (if there were no listed options) because UPS didn't deliver to us. IIRC, USPS had a 20-lb limit on parcels at the time, so larger stuff would have to ship by UPS.
Dad sometimes would hand me a tool or parts catalog with a couple of pages and items marked, and have me fill out the order form. I think that was a test more than anything else, but I smile when I remember it.
I can see where if you lived in town, and tended to only need stuff that was available at local stores and businesses, you wouldn't have needed to mail order stuff much, and you may not have needed or wanted a credit card. Also, it's pretty obvious that the traditional walking mailman wouldn't have capacity to carry parcels in his shoulder pack or on his tiny cart. I know my grandparents, who lived in the city, occasionally got parcels, and they were delivered by a driver, not by their regular mailman.
Just for curiosity, I looked up the history of the parcel post. USPS has had it since since 1913, although they were accepting smaller packages long before that. Sears was shipping 30,000 orders a day in the early 1900s.
A particularly interesting quote: "In 1903 Sears claimed that “one-fourth of the entire population of the
United States secures some of their goods from the Chicago Mail Order Houses,” which by then were
receiving up to 30,000 orders daily."