First, to make sure you are aware that when running spellcheck, Word checks
all the main text, then returns and checks the notes, so it may look like it
is skipping it, but it isn't.
If that's not it, then the note text may be set to No Proofing, or the wrong
language. The text has to have a language assigned for Word to spellcheck
it.
To test this, with the cursor in a word that isn't being checked, go to
Tools | Language and see what comes up selected.
To fix this, Modify the Footnote Text style to have a language. One way:
Format | Style, select Footnote Text in left column, click Modify, click
"add to template" but not "automatically update", use the dropdown Format
menu to set a Language.
DM