It's technically correct that we need some animal products. The big one I can think of is crude oil. It's from animals long dead and also used to make plastics. Not sure if we count insects as animal products but the work of bees helps pollinate our crops. We don't eat the bees but we do take advantage of them and breed them to go around and pollinate crops.
IIRC that's mostly from plants and algae, and definitely can be substituted with things made purely from plants and algae.
However, I was replying to someone who said specifically "meat", and crude oil isn't meat.
> Not sure if we count insects as animal products but the work of bees helps pollinate our crops. We don't eat the bees but we do take advantage of them and breed them to go around and pollinate crops.
If you're that expansive, humans are also animals, and we also rely on each other's labour.