By the time of widespread home delivery, much of the ice was being made by artificial refrigeration. It’s just that the systems of the late 19th century were industrial-scale, not suited for home appliances.

Earlier in the 19th century, however, ice harvesting was a large industry. The European industry was concentrated in Norway. In North America, lakes in Upstate New York, Minnesota, north of Chicago, and throughout New England that had clear, clean water were carefully watched in winter. When they froze a foot thick, the surface ice was cut into large blocks that were stored in nearby sheds, covered with hay or sawdust for insulation, and shipped as needed by barge to Philadelphia and New York, in insulated boxcars (earlier, presumably in wagons), and even ships, to reach the tropics.

You should check out the book, The Frozen Water Trade. It goes into fairly concise detail about the ice industry before refrigeration and before ice making machines existed. In brief: ponds in New England and Canada (and other northerly coastal areas) froze in the winter and were then cut into large chunks. The chunks were packed into barns with sawdust and/or hay until they were needed in the spring and summer months. They would then be shipped via horse cart, train or boat to their final destination. In fact, they used to pack entire wooden ships full of ice and sawdust and ship it as far as India.

It's worth adding, with regard to the New York ice trade in particular, that ice was a crucially important product for most families during the summer months. Before the advent of refrigeration, it was the only way of keeping foods fresh and of preserving some medicines. This encouraged both politicians and criminals to participate in the trade and seek to construct potentially lucrative monopolies.

Thus the fall of the New York mayor Robert Anderson van Wyk (after whom the Van Wyk expressway was named) was occasioned when the New York World discovered that his administration had granted an effective monopoly on the supply of ice to the city to the American Ice Company, which was the only group that had the right to land ice at NY harbor piers – and that American Ice was planning to take advantage of its position by doubling the price from 30 cent to 60 cents per hundred pounds, putting it beyond the reach of some families. Further investigation revealed that Van Wyk (whose salary as mayor was $15,000) owned $680,000 of American Ice stock, which he did not appear to have paid for.

On a more local level, Vincenzo Terranova, who was the half brother of Giuseppe Morello, the first boss of one of New York's Mafia families, worked for many years running an ice delivery operation in the Italian districts of New York. The presumption is that the Mafia were exploiting demand for ice to carve a local monopoly for themselves, as they did with other products that Italian immigrants could not do without, such as artichokes.