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.
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.