S.D. Codified Laws § 12-6-56
If a vacancy occurs by reason of death or withdrawal after a primary election, a party candidate for public office may be chosen if a meeting of the appropriate party central committee is held and the results are certified to the appropriate official within the times prescribed by § 12-8-6.
If the vacancy is a party candidate for presidential elector or statewide office, the vacancy shall be filled by the State Party Central Committee. If the vacancy is a party candidate for public office other than presidential elector or statewide office, the vacancy shall be filled by a vote of county party central committee members in attendance who reside in the affected district.
The certification of the results shall be signed by the person responsible for calling the meeting under § 12-6-57 or the person designated to conduct the meeting under § 12-6-57.
SDCL 12-6-56