Hello! I'm a former zookeeper and I have worked with all kinds of waterfowl, including just about every duck species you can imagine, in the past.
It is true that ducks tend to live in swampy or riparian areas, with lots of mud underfoot...but in the wild, they are able to move around freely and stand on all kids of surfaces throughout the day. In an enclosure, they're limited to one or maybe two standing surfaces due to their dabbling behavior ("chomping" in shallow water/sloppy mud, which tends to quickly make more mud...and more mud...and more. I'm sure you've seen this first-hand!)
As you noted, adding straw and other vegetative matter to the muddy areas will most likely cause mold, which can be a health issue for them. (But they will enjoy playing with the occasional load of straw, so offer it now and then.) Instead, I'm going to recommend a not-very-organic solution. We used it all the time with captive waterfowl in zoos. Get some of that green plastic Astroturf-like "fake grass" stuff from the hardware store. They sell it in big rolls, and you can have them cut pieces to your specific size needs. Get two or three pieces at least, which you can rotate out of their enclosure every day or every other day. You can put down a mat in the most relatively dry spot you can find (maybe under a heat lamp, if you have one) and the ducks will stand on it. The little plastic "grass blades" are sturdy enough to keep their feet up out of the damp. This will give them some relief from standing in wet areas all day, every day, and it should allow any abrasions on their feet to heal so you don't get the infections that cause bumblefoot.
The "grass" mats are easy to take care of--you can just spray them with a high-pressure hose (your thumb over the nozzle, old zoo trick!) to remove feces and other debris. Pick the soaking-wet mat up and beat it against a sturdy structure of some kind (the side of a garden shed, maybe.) that will get most of the excess water out and knock out any debris the hose couldn't get. Then hang the wet mats over a fence or similar structure to dry in the sun. The UV rays from sun exposure will kill any excess bacteria or viruses hanging around in the mats. Let them dry thoroughly between uses, so you don't expose the ducks' feet to more bacteria--especially important if they have active infections or breaks in the skin. This hosing/sun-drying process is why you should get a few mats and rotate their use regularly.
Good luck!
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