According to data published by the US Deparment of Housing and Urban Development (hud.gov), the total number of homeless US veterans recorded on a single night in January of 2022 was:
On their website they describe what the data shows as:
The data show that on a single night in January 2022, there were 33,136 Veterans who were experiencing homelessness in the United States – down from 37,252 in 2020. Overall, this represents a 55.3% reduction in Veterans experiencing homelessness since 2010.
A screenshot from their page is below. A complete PDF of the page from the site is in the evidence links below.