While I can't provide an answer to how "common" archery was in the USA in 1900, I can contextualise the rise of organised competition in that time period.

Of particular note is the work done by brothers J. Will & Maurice H. Thompson after the Civil War, in which Confederate soldiers were not allowed to own firearms, necessitating the use of the bow as a hunting tool. The Thompson brothers would publish two notable texts, How to Train in Archery and The Witchery of Archery - the former being a comprehensive text on competition shooting using the standard York round, the latter being a detailed manual on bowhunting and fieldcraft.

The Thompson brothers founded the National Archery Association (NAA) in 1879, which continues today as USA Archery, the national governing body for sport archery.

The 1904 Olympic Games in St Louis was host to an all-American event, otherwise formally the 26th Grand Annual Target Meeting of the NAA, with 23 men and 6 women competing across all events.

Notably in the 1920s, there was a revival of interest in Native American archery through the work of Saxton Pope, relating his experiences with Ishi, the last Yana Indian.

In brief, while I doubt we can come up with exact numbers to ascertain how common archery was, we can certainly say that archery was promoted as a form of hunting and sport, given the foundations of significant individuals and organisations during this time.

Further Reading:

• The Archery Library
• Olympedia
• History of Archery and the NAA, Washington Post
• Robert J. Rhode, History of the National Archery Association of the United States of America, 1879-1945