Not to my knowledge. PLA has its utilities but it isn't as durable. I would be shocked if most vegans avoid plastic. 

That said, I like to minimize plastic usage but not because of the oil connection or the animal connection.  Plastic is an amazing invention. 

I'm aware of PLA as a plant sugar-based plastic, but it looks like that has less than 1% marketshare in product packaging globally.

Are there any other more popular plastics that don't contain any animal products?

There are plant based plastics but you ask a good question.