BILLUND, Denmark—Lego A/S has spent the past seven years trying to make its blocks with plastic derived from plants, but it keeps hitting brick walls.

Lego tried making pieces from corn, but they were too soft. Its wheat-based bricks didn’t absorb color evenly or have the requisite shine. Bricks made from other materials proved too hard to pull apart, broke or had what executives call “creep,” when bricks lose their grip and collapse.

“It’s...