Across Southeast Asia, governments are seeking Russian oil and gas to ease fuel shortages triggered by the continuing chokehold on the Strait of Hormuz, a crisis that has sent energy prices higher and forced import-dependent countries to look beyond their usual suppliers.

But analysts say the scramble for Russian fuel also raises a bigger question for the region: whether Moscow can turn a short-term role as an emergency energy supplier into longer-term influence there.

Member states of the...