Thank you for all of your cute basket suggestions! I went with plastic lace baskets for toy storage. I had completely forgotten about them, but they were all over the design blogs when they first came out as the Carrie bike baskets. Victoria uses one for file storage, Joy has one for scraps in her studio, and now I can tell you that they also make excellent little toy bins.
p.s. Have I ever mentioned how much I love my vintage side tables? They’re the best. They hold remotes in the drawers, magazines and the new toy baskets underneath, and they have ample surface space up top. And they were only $10 each.