In Skyrim for the Nintendo Switch, how do you obtain the pets for sale quest?
By "Pets for Sale" quest, I'm assuming you mean the "Pets of Skyrim" quest (a quest to obtain multiple pets).
The Pets of Skyrim quest is added by the Pets of Skyrim Creation Club DLC. You need to buy or upgrade to the Skyrim Anniversary Edition for the Switch (which contains Pets of Skyrim, among other Creation Club DLCs). It can not be played on the Switch if you don't have the Anniversary Edition.
According to the UESP entry for the Pets of Skyrim quest, the first step in obtaining the five pets is to
Read the For Sale notice at The Bannered Mare.
The notice will give you the first location to travel to.