Chickens, cows, paid leave, even a million-dollar apartment, these are the prizes on offer from Thailand to Indonesia to Hong Kong for those who get vaccinated against Covid-19 as nations across Asia combat vaccine hesitancy.

As more transmissible variants of the virus have driven a surge in cases in Southeast Asia in recent weeks, authorities have started offering chances in lucky draws for vaccinations.

In Thailand’s Mae Chaem district in northern Chiang Mai, where most residents are cattle farmers, authorities launched...