It would seem that widespread telephones would be necessary since you need a way to call ahead and if you went out to tell the restaurant what you wanted in person you might as well eat it there. But then again, there were other messaging systems and advanced postal systems that predated the widespread adoption of the telephone.
We know that butchers in Paris delivered their wares as early as the late 1300s; there is a list of them in the Good Wife's Guide. But the first record of delivered meals was from Korea in 1768. The Ijaenango, a 50+ year diary kept by Hwang Yun-seok, recorded that he ordered naengmyeon and ate it with his friends. But naengmyeon is a cold dish, and you asked about hot food.
We see hot food delivery appearing at about the same time in both Korea and India. In Korea, haejangguk, a hot soup, was delivered in the late Joseon period, roughly 1880-1890. (This article is in Korean; Google Translate should be helpful.)
At the same time, the first dabbawalas started up in Mumbai. In 1890, Mahadeo Havaji Bachche started a group of young men who delivered hot lunches to clients. Today there are at least 5000 members of the dabbawala union. For more reading on dabbawalas, see the Harvard Business School case study The Dabbawala System: On-Time Delivery, Every Time .
In the US, the first delivery was Chinese food in about 1920. The Kin-Chu Cafe delivered from 11am to 1am; Time magazine cites that clipping as from 1922.
If you'll excuse me, I'm pretty hungry now...
