Word normally does spellcheck headers and footers. If your headers and
footers are not getting spellchecked, then it's probably because the text is
marked as a language for which you do not have proofing tools, or the text is
marked as "no proofing", or the paragraph style is marked as "no proofing".
Try selecting the text in the header, then click on Tools | Language | Set
Language | <choose your language> | uncheck the "Do not check spelling or
grammar" and "Detect language automatically" boxes | Default | OK. Do the
same thing for the footer. Close the footer, and press F7 to rerun
spellcheck.