Are there any publicly available faces database where each face is annotated with the person's height and weight?
There is the MORPH dataset but it is not publicly available.
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If you're willing to get your hands dirty, there are plenty of websites out there that post mugshots of arrested individuals. Many of these come with height and weight attributes. With a little scraping you could have as many photos as you could possibly want. You might have to deal with watermarks and artifact removal though.
As an example, here is a collection of mugshots from Polk County Iowa. It even allows filtering by height, weight, age and gender.