How can I send a conditional email to edit one question that does not show up until the respondent receives that email?

If the respondent selects a particular multiple choice option on the original survey, then I would like them to receive an email upon submission of the form asking them to fill out one question (because they have to approve the contract upon completion of the work in this case).

This guide should help you configure the conditional logic email: https://www.jotform.com/help/167-How-to-send-to-a-specific-email-address-based-on-a-selection.

Note that you need to create the email template and add your instruction there about the other field. Also, is this case, you need to add the Edit Link in the email template so they users can edit and re-submit their entry. To add the edit link, follow this guide: https://www.jotform.com/help/40-How-to-Let-Users-Update-Their-Form-Submissions-at-a-Later-Date. 

Let us know if you need further assistance.