Here are two very rough ways to get an answer:
1. There are 300,000,000 people in the US, and maybe 1 in 100 (at most) has a telescope. Thus there are 3 million amateur astronomers in the US.
2. There are 50 states in the US, and each one has, say 4 astronomy clubs, with perhaps 150 members each. That's 30,000 amateurs.
While these numbers seem reasonable, the overall result is pretty poorly constrained. Can we do better?
Another way to estimate the number is from magazine circulation. A bit of web-searching reveals that Sky & Telescope and Astronomy magazines have circulations around 100,000, so double or triple that (since not all amateurs subscribe) to yield 300,000. By the way, that's the geometric mean of the two estimates above. Does 1 in 1000 people owning a telescope seem right to you? Is that close enough?
Probably a more pertinent question at this point is just what is a amateur astronomer? Certainly arm-chair enthusiasts count, as do those joining a club or purchasing a telescope.
How would you recognize a bona fide star-gazer if you saw one? Knowledge of the sky, solar system and constellations? Size of telescope and skill at using it under the starry sky? Total dollar investment in optics electronics and other accessories? Willingness to share what they know with those less familiar with the sky?