Wow, still getting replies!  (OP here.).  I tried to find the soft and stable, and also some thin polypropylene locally, and came up empty.  I ended up using some thin decorative foam, and it worked okay.  For the next project, I'll try to get something else as part of a larger order, so the shipping cost doesn't swamp the cost of the materials.  Thanks for taking the time to answer!

Hi, I'm new to MYOG.  My first attempt at a top tube bag came out okay, but would be better with stiffer side panels.  I thought something like 1/8" closed cell foam sandwiched between 2 layers of fabric would work, but I'm having trouble finding the right material.  What do you use, and where do you get it?  Do you have any related tips you can share?  Thanks.