In a sefer called "Toldot Chag Simchat Torah" by Avraham Yaari, He expalins the origins of many Simchat Torah customs. (sorry, can't find a copy online)
It seems like he may mention* the custom in the book, but I'm not sure since this blogged timeline of Simchat Torah, made from the book, adds the custom of tying talitot in the late 20th century and the book was published in 1964. So perhaps it came about sometime after 1964:
from the blog timeline, not the sefer:
Late 20th century...Yaari, p.85 mentions earlier customs of having three or four chatanim,
but not related to maftir. I also remember as a child that there was a
custom for the children to tie the talitot of the adults, but I have
not seen this done in years.
As I don't have a copy of the sefer, I can't check if he actually mentions it. However the oldest reference to the widespread general Shtick that went on during Simchat Torah is possibly from the Diary of Samuel Pepys (a famous 17th century English diarist). On Simchat Torah, Wednesday 14 October 1663, he visited a shule for the first time and records what he saw.
"But, Lord! to see the disorder, laughing, sporting, and no attention,
but confusion in all their service, more like brutes than people
knowing the true God, would make a man forswear ever seeing them more
and indeed I never did see so much, or could have imagined there had
been any religion in the whole world so absurdly performed as this. "
via & via
*Edit: this blog supports that Avraham Yaari's Toldot Chag Simchat Torah mentions it.
and of allowing the young men to take over proceedings, including the
old shtick of tying people's tallitot together, stealing food from
ovens, etc. etc. etc. This was all very widespread but apparently
Salonika was particularly known for letting the service become jokey.