My refrigerator has a molded 1” wide edging that runs along the sides of the unit. About half way up the walls.
Does your icebox have any sort of feature such as this? If not you may be able to adhere a piece of PVC to the walls, using PVC cement, If the ice box is PVC itself. There are other PVC cements designed to glue PVC to vinyl etc.
Here are some pictures of ours for ideas.
The 4 racks on the left hold a bottle of beer at just the right angle to reach. The larger ones hold a wide bottle of wine or 2 liter bottles of club soda.
The shelves are removable to allow access to the bottom of the refrigerator and taller items.
The white shelf under the freezer part is removable as well.
Some ideas for you to look at.