For most non-food shopping needs, you’ll honestly have better luck driving to Kumamoto or Nobeoka, as there aren’t any malls or large department stores. However, for your day-to-day needs, the following stores will tide you over.

Specialty Center Gokase 特産センターごかせ
Located on Highway 218 near the border between Miyazaki and Kumamoto Prefectures, this shop features local fresh vegetable and specialty products from Gokase Town. The center also has a restaurant featuring menu items made with Gokase ingredients.
Location: 特産センターごかせ (map)
98-1 Sankasho, 西臼杵郡西臼杵郡五ヶ瀬町 Nishiusuki District, Miyazaki Prefecture 882-1203
TEL: 0982-82-1400

ACOOP Aコープ五ヶ瀬店
Smaller than the Takachiho store, with an equally smaller selection, but more convenient.
Location: 高千穂地区農業協同組合五ヶ瀬支所Aコープ五ヶ瀬店 (map)
〒882-1203 宮崎県西臼杵郡五ヶ瀬町大字三ヶ所10721−2
TEL: 0982-82-1124

DeoDeo Yamato デオデオ山都店 
A small electronics store located over the border in Kumamoto. Not the best selection or price, but close by.
Location: デオデオ山都店 (map)
794-8 Mamihara, Yamato-chō, Kamimashiki-gun, Kumamoto-ken 861-3901
TEL: 0967-83-1755

Yamazaki Yショップ五ヶ瀬
A small convenience store, and the only store in Gokase that is open after 8pm. Shop there if you need anything in Gokase at night.
Location: Yショップ五ヶ瀬 (map)
2137 Sankasho, Gokase, Nishiusuki District, Miyazaki Prefecture 882-1203
TEL: 0982-82-0269

Actually in Kumamoto, but right by the border.