You've already explained why you cannot have a truckload of wood chips dumped into your run, which would have been my first suggestion

Here are a few other things I thought of. Maybe one of them will help, or will inspire some other idea.
You could have a truck full of wood chips dumped on your driveway, then move them to the run with a wheelbarrow. (A lot more work than having them dumped directly in the run, and depending on the layout of your property may not be feasible at all.)
Fall is coming. Dead leaves can help with a muddy run. They may pack down in layers, except the top layer that the chickens keep scratched up, but that may not matter. Of course you can rake leaves on your own property, or offer to rake leaves for someone else. But if people in your neighborhood rake their leaves and sit them out in bags for the trash company, you could take some of those bags home to dump in your run. Or offer to "help" a local church or business by hauling off the bags of leaves that someone rakes off their property.
Bales of hay or straw might help, at least temporarily. (Either spread around, or just sitting there still baled.)
Dumping in more dirt might help, by raising the ground level a bit, so the water prefers to run somewhere else.
I've read of people laying pallets or logs or stumps in a wet chicken run, so the chickens can sit up on something dry. (This does not prevent bad smells from a muddy run, so not a complete solution.)