Your Lowe's should have everything, ours does. Well with exception of the rolled vinyl sheets. I can get that at ACE hardware, they sell it in 3 different thicknesses, cut to order (and sold by the yard). But again here many materials can be used, as long as it's water proof, flat, and can be glued to the Styrofoam. They also sell at Lowe's and Home Depot rolls of the 6ml thick plastic (both clear and black). It's similar to a plastic drop cloth but much thicker, and sold in larger rolls. I believe it will be in the home insulation department. I plan to use the 100 foot by 20 foot clear 6ml roll as a covering for a planed greenhouse. However I would use the black for inside a cooling box, it wont show stains and glue spots through it.
Another perfect material for this is called FST or FDS board (I think). It's what they use as shower walls (even commercial kitchen walls, inside walk in refrigerators etc.), and it's very durable and easy to clean. But it isn't cheep, runs about $30 to $35 for a 4x8 sheet. Another almost as durable but less expensive option is using a pond liner. It's a very thick black plastic sheet that's very durable, and comes in various sizes. It's usually found in the nursery department, and is the base for man made ponds and waterfalls in peoples backyards. They dig the hole, then spread and form the liner to the whole size.
The 2 inch Styrofoam sheets will be in the home insulation section at Lowe's, Home Depot etc.. Also I forgot I had the picture of the foamboard adhesive, that should be with all the other adhesives like liquid nails. But some of the solvents in those will dissolve Styrofoam, the foamboard adhesive is designed for gluing Styrofoam to wood frames, and other construction materials.
Ya, it can be amazing how much water the plants drink, especially the bigger they get. We don't really get to see/measure that when growing in soil.