BEIJING/SHANGHAI—China’s “common prosperity” crackdown has turned a harsh spotlight on the country’s massive livestream e-commerce business—underlining the fragility of a surging sales channel that some of the world’s biggest brands have come to rely on. More than 100 million followers of Viya, dubbed the country’s “queen of livestreaming” by the Chinese media and public, awoke on Tuesday to find her e-commerce and social media accounts shut down after news that she had been fined more than $200 million for tax evasion. The rise of celebrities partnering with brands from L’Oreal to Unilever and Adidas to sell consumer goods in live online streams has seen the sector billow. Consultancy McKinsey expects the trade in the world’s second-biggest economy to be worth $423 billion next year—more than double estimates for 2020, and bigger than the economies of countries like Norway and Ireland. But it has also led to an awkward tango for …