This is absolutely false. Every three years, by law, NYC does a housing and vacancy survey of tens of thousands of apartments (the most recent survey was in 2017). The raw data is here[0], and selected findings are here[1]. You can see in in the selected findings that the vacancy rate in NYC in 2017 was 3.63%, which is INCREDIBLY low. A "healthy" vacancy rate is closer to 7% or 8%[2].

[0] https://www.census.gov/programs-surveys/nychvs.html

[1] https://www1.nyc.gov/assets/hpd/downloads/pdfs/about/2017-hv...

[2] https://www.strongtowns.org/journal/2020/8/30/what-vacancy-r...