I am worried about the goats' pen getting muddy so I was thinking of having a tree trimmer drop off a load of wood chips. Are there any problems with this plan? Can I use the mulch as bedding in their shed too?
Organic material breaks down and holds water. That is why gardeners use mulch...to hold in moisture. They have to continually add more as it breaks down. Wonderful in a garden, but a sticky mud pit where animals are concerned. If their pen tends to get wet, you might build it up with sand, but any plant material will ultimately create mud.
My pasture and paddock are on sand (we were probably the bottom of a river or ocean a million years ago here) and it drains very quickly. After eight years of hay racks and bedding getting tracked out of stalls, that area is very muddy for days after rain. Until I can get it scraped out and the sand replaced, I put blocks for the goats to walk across, as you know how they feel about water!
In the long term (especially on poorly draining soil) I am 100% with what free said above.
However I would point out that sometimes a load of tree chippings or hogfuel can be an excellent solution to a temporary severe-mudpit problem. Yes, it will make things worse in the long run (like, seasons or years) unless you are on very free-draining soil (under which conditions it is quite unusual to *have* a mud problem in teh first place, but it can happen). But if your horses are up to their knees and hocks in squelch, the next few months may be your big priority and you figure you'll deal with the bigger picture when things dry out enough to get the tractor in there
Too, I know people in the pacific NW who get in a new load of hogfuel every year for their horses' winter paddock, and have done this for like twenty years, and no longterm problem has developed. I believe this is because they are bringing in so much new material so frequently. I certainly do not guarantee it will work for everyone, but, it does work for *some* people.
The big thing when using tree chippings or hogfuel for mud-amelioration in horse paddocks, and I can't see why it'd be much different for goats, is to make real sure of your source. Not only does it have to NOT contain large dangerous spikes of wood, or miscellaneous wire-and-metal-fragments-and-plastic-bags, it also needs to be free enough of green leaves that it does not get all moldy and hot. So what you ideally want is something chipped from DEAD trees. Failing that, from evergreens or from lost-all-their-leaves-already deciduous trees.