SHANGHAI—China’s insurance regulatory agency Friday took control of hard-charging, acquisitive Anbang Insurance Group Co., saying the action is needed to avoid a collapse of the firm following suspected illegal activity and the downfall of its once-highflying chairman.

Anbang was among a handful of privately run, fast-growing Chinese firms that appeared to enjoy unusual levels of political support to borrow money and build global footprints. For instance, it owns New York’s iconic Waldorf Astoria hotel. Many of those companies...