Does a twenty-rak'ah prayer vowed for committing a sin suffice for a missed Sunnah of Dhuhr, or must the Sunnah of Dhuhr be made up afterwards?

This vow is a vow of contention (nathr lajaj), and what is obligatory concerning it is one of two things: fulfilling it or offering an expiation for an oath. Since you fulfilled your vow and prayed the twenty rak'ahs, there is nothing upon you. As for the Dhuhr sunnah prayer that you missed, making it up is permissible, and it is recommended to make it up at any time, though hastening to make it up is preferable.