I was sick on the day of the PSAT and wasn’t able to receive an alternative exam. After emailing the College Board, they said I could use my SAT score for alternate entry into National Merit. I was wondering if I plan on taking the SAT to get into NM, would I have to send that score to colleges (whereas PSAT scores don’t get sent)?
The SAT you will be taking for alternate entry shouldn’t have any restrictions on it - so yes, you could end up sending it to colleges as well as NMSC. In fact, if you score well you might WANT to send it 
However, unless you are applying to Georgetown (requires all scores), you can always send your ACTs or chose which best tests to send (ACT vs. SAT). Check your colleges’ testing policies on their websites.