Turns out it was trivially easy to make a soldering iron holder out of a couple pieces of scrap wood, some screws, and a "spring grip" -- one of those spring grabby things you can push a broom handle into to hold it in place against a wall. Tools: saw, screwdriver, drill.
Step 1: Cut the wood
You do want to assemble the finished tool so it holds your soldering iron at a good working angle, but doesn't obstruct the cord of it. I found that 45 degrees was too much, and used an angle closer to 30 degrees. The base only needs to be big enough to screw the triangle to it, and have some stick out for the clamp to attach to.