The old man is operating an illegal business in a run-down building in New York. He does not advertise and obviously keeps a low profile. He depends for customers on recommendations of others who have been satisfied with the effects of his potions, both the original love potion and later with the "chaser." We know that Alan Austen has been given the old man's name and address in confidence.
Alan, without a word, handed him the card he had been given.
There probably is no law against selling love potions, but there must certainly be laws against selling undetectable poisons to men who want to dispose of their wives. As Emerson wrote:
If you build a better mousetrap the world will beat a path to your door.
Some things in stories have to be taken as "givens." We have to accept the author's specification that there is actually an old man who deals in love potions and poisons. The French call this a donnée, which means a premise the reader must accept if he wants to appreciate the story. There are many stories with farfetched assumptions, or données, such as the assumption that a man like General Zaroff actually hunts humans on Ship Trap Island in "The Most Dangerous Game."
The author of "The Chaser" made many farfetched assumptions, and they provide delight for his fans. In one of his stories collected in Fancies and Goodnights, Collier tells about a poet who manages to survive in the big city by posing as a mannequin in a department store during business hours and being free to roam the store and write his poetry by night. He finds out that all the other mannequins are people doing the same thing.