Once Mongol forces during the reign of Chinggis' grandson Kublai invaded Vietnam in the late 13th century, they would have had a good idea of them.

But I would imagine there was a general idea that a group called the Mongols was conquering places: the Mongols had been at war with the Sung Dynasty since the 1230s (I don't have my notes with me at the moment so exact dates escape me) and had made their first forays into China with the 1209 invasion of the Tangut kingdom, vassalizing them in 1210, then attacking the Jin in 1211, destroying the Jin capital Zhongdu (modern Beijing) in 1215. The Jin were ultimately destroyed in 1234.

Throughout this period, the Sung were Well aware of the Mongols, although whether they ever anticipated the Mongols would go after them is questionable.

The Sung, as the major economic Power of east Asia, had extensive trade contacts by land and sea with its neighbours, and while I can't say for certain I would bet good Money the Sung were one of the main trading Partners of the Vietnamese Kingdoms. So, Sung ships/merchants would have certainly brought information about what was happening, and as the Sung-Mongol wars picked up in the 1250s (in which time Tibet and significantly, Yunnan, were conquered by the Mongols) the Vietnamese were almost certainly aware of what was happening.

Now, how much they knew is debatable: When I get a chance later today I'll do some reading and See if I can find anything more Out on this. I doubt they knew much during Chinggis Khan's lifetime, and may have not known his Name specifically, and they certainly did not know the scale of the conquests, but by the 1230s they were likely hearing rumours from Sung traders that Northern China was being overrun, and certainly by mid-century they knew much more as the Mongols worked their way south.

I'll update this later when I get a chance to do some reading later today: Most of this is from the Cambridge History of China series on foreign dynasty's in China 907-1368, and I'll look through their chapter on Kublai and See what more Fun facts we can dig up.