Qandala lies 75km (46 miles) from the commercial port city of Bosaso, in the north-eastern region of Puntland.
Qandala district commissioner Ahmed Khurshe confirmed the fall of the town to IS to journalists, saying that some residents, including himself, had fled.
Reports suggest the fighters set up a military camp inside Qandala, which itself is a coastal town.
Puntland's regional administration is holding an emergency meeting to discuss the situation.
A year ago, a group of al-Shabab fighters in Puntland pledged allegiance to IS.
The breakaway faction is led by Abdiqadir Mumin, who the US put on a sanctions list last month because of his terrorism activities.