Can anyone tell me how many otolaryngologists (ear, nose and throat
doctors) and audiologists are practicing in the U.S.?

Hello.

There are an estimated 10,000 to 11,000 otolaryngologists in the U.S.

The American Academy of Otolaryngology has approximately 10,000 members.
See: AAO: About the Academy
http://www.entnet.org/about/index.cfm

"There are only 11,000 otolaryngologists in the entire country, and
only 300 are pediatric subspecialists."
source: OMEW.com, Dr. Michael E. Pichichero 
http://www.omew.com/Tymp_article.htm

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There are an estimated 10,000 to 12,000 audiologists in the U.S.

"There are approximately 10,000 audiologists in the United States."
source: HearAOK.com
http://www.hearaok.com/audiologists/audiologists.html

"With approximately 12,000 audiologists in the country, every
audiologist potentially could have 800 more patients to work with."
source: Advance for Audiologists Newsmagazine
http://www.advanceforaud.com/common/editorial/editorial.aspx?CC=6258

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search strategy:
"otolaryngologists in the" country
American Academy Otolaryngology
"audiologists in the united states"
"audiologist in the country"

I hope this helps.

Thanks so much for your quick response!