Taoism is not generally known as a proselytizing religion these days, but was there ever a time when Taoist missionaries were found?
Taoism gained much popularity by the end of the Southern Song period.
When the Mongol Empire started invading, they had the habit of massacring the whole population after a city was sacked. Taoist missionary Qiu Chuji visited Genghis Khan and managed to convince him not to commit any more massacres after battle.
Generally speaking, there were very few Taoist missionaries in Chinese history.
The most famous Taoist missionary would be Zhang Jue from the Three Kingdoms era:
From the Chinese wiki: In AD 184, plague spread throughout China. Thousands of people were cured after drinking Zhang's ash water. He sent 8 major missionaries and his followers covered around 2/3 of China. They then became the major army of the Yellow Turban Rebellion. This rebellion led to the end of the Han Dynasty.
Yes, and there still are.
Taoism is generally not known as a proselytizing religion because most Westerners don't read Chinese and the ones who do, especially early on, mythologized and orientalized it. The Confucians had no such qualms and tried to get it banned as a noxious superstition by the Rites Bureau whenever they got the chance. The Chinese Taoist Association isn't as aggressive today because it's run to harmonize its activities with the atheist CCP, but historically China has had plenty of popular religious movements and the only way to pretend they weren't Taoist is to No-True-Scotsman them all. (The English catch-all for these NTS movements and ideas is 'Chinese Folk Religion'.)
Val's answer is right that the plagues, floods, and wars that ended the Han Dynasty seems to have been a fertile time. Zhang Jue and his Yellow Turbans were dramatic and influential but short-lived; Zhang Lu (no relation) and his Celestial Masters following the Way of the Five Pecks of Rice had epiphanies of Laozi, medical miracles, and tried to establish a holy land for their chosen people. Their ideas shaped most subsequent Taoist movements.