The short answer to this question is “it depends.”
Driving under the influence (DUI) of alcohol or drugs is considered a criminal offense in many places. That also means that it may show up on a criminal background check or on a motor vehicle record (MVR) check.
Whether or how a DUI shows up in a background check report can depend on whether there has been a conviction for the DUI or if it is still a pending case. Criminal searches can show pending cases. DUIs are less likely to appear on an MVR if the case is still pending.