I'm liking Bob's thoughts about milk cartons, doubling (tripling) it over if you want more stiffness.
I often dumpster-dive at the recycling center, especially in the HDPE bin (milk jugs). Other products also come in HDPE but are thicker gauge - detergent, shampoos, coffee, etc. Cutting a strip from one of those would give you stiffer material.
For really thick, but cheap HDPE sheet, I always keep a few of these around:
They retail for $6-10 but are half off come Spring. Or $1 at a thrift store. For a quick, long strip; roll it tightly, tape the roll securely and use a chop saw to cut off a section of roll. It could 2", 1" or 1/2" thick. The cut edge will be rough, but can be smoothed easily with sand paper or those Stanley Surform rasp-like tools you pull towards you:
Or mark along a straight edge with a sharpie and then cut your strip off with shop shears, or some heavy-duty scissors.