Unfortunately, I can't reproduce this. When the screenshot is working for PDF, it is always working for WORD output.
What exactly is the HTML content you have in your book ? Is this only HTML widgets ?
The screenshot will only be done automatically for supported format (widgets and shinyapps mainly). If you are using other type of HTML output (maybe a HTML table ?), there will be no screenshot.
Can you identify the type of content (HTML Widgets I guess) you are using ? It seems that one is not getting screenshoted while rendering to word.
No that you can force screenshot by setting screenshot.force to TRUE in a chunk that would require it. But it should be automatically be set to TRUE for chunks outputing object of class htmlwidget.
Something ike setting the knitr option screenshot.force = knitr::pandoc_to("docx") like could maybe force the screenshot to happen, but you would need to know on which chunk to put it.
anyway, this is supposed to work. Specifically if it works for PDF, it should work for WORD.
Without more details, it will be difficult to help more. You'll need to look closer in your book content to find what is creating this.
Hope you'll find it!