I am a landscaper & have in the past, & am right now installing commercial playground safety surfacing. So I am doing thousands of dollars worth of drainage work for some sites, before the playground mulch goes in. Wood is a bidegradable product, so it will break down over time, and will need to be replenished over time, just like the mulch you may have in any shrub beds etc. (generally every 1-3 years). Having your wood sit in a puddle will break it down a lot faster & create the other problems mentioned, and possibly make the kids get messier than they already can on their own? (Like they need help right? /forums/images/graemlins/grin.gif)
So as mentioned, you want to do something to get that water out of there. Here are a couple of the cheaper, easier solutions for you. Regrade with a slope in the bottom to direct all of the water to one corner or side, then install drain tile in a way to get it out. Wrap the drain pipe in a layer of stone & geotextile fabric. Another option done often is the bottom will be excavated deeper than needed to allow for a layer of stone to hold the standing water until it filters down into the existing soil (4"-6' or more as needed). With your soil I'd suggest the first option. Either way, you want to put a geotextile fabric layer between each different material (thicker, better quality weed block fabrics will also work & be cheaper). Between the existing soil and drainage stone, then between stone and playground safety surfacing. This helps keep the soil from mixing into the stone or into the mulch, etc. & will help keep the mulch from breaking down quicker by mixing with the soil.
As far as the mulch goes, buy Playground Safety Surface Mulch. There are several companies manufacturing this and should be available in most areas if you search for it. It is designed & built to be safe for kids to fall on, and is also cleaner than regular mulches. Follow their instructions as far as how deep it should be installed in reference to the heights of the playground equipment being used.