Yes, private network computer reservation systems were in widespread use in the airline industry prior to the 1980s:
A computer reservations system or central reservation system (CRS) is a computerizedsystem used to store and retrieve information and conduct transactions related to air travel, hotels, car rental, or activities. Originally designed and operated by airlines, CRSes were later extended for the use of travel agencies. Major CRS operations that book and sell tickets for multiple airlines are known as Global Distribution System (GDS).
The first remote CRS was up by 63!.
In 1953, Trans-Canada Airlines (TCA) started investigating a computer-based system with remote terminals, testing one design on the University of Toronto's Manchester Mark 1machine that summer. Though successful, the researchers found that input and output was a major problem. Ferranti Canada became involved in the project and suggested a new system using punched cards and a transistorized computer in place of the unreliable tube-based Mark I. The resulting system, ReserVec, started operation in 1962, and took over all booking operations in January 1963. Terminals were placed in all of TCA's ticketing offices, allowing all queries and bookings to complete in about one second with no remote operators needed.
The public Internet was also up by that year, 100,000 computers strong. But Airlines used private networks.
And that computer thing is called a terminal.
Also anyone over fifty can confirm this was real.