Be careful with using sod - you need to verify it does not have any chemicals such as fertilizers added to it. Sod is often doused in all sorts of stuff to make it grow.
Also rather expensive for something that the chickens will tear up and destroy in no time flat. If I give my chickens "sod"...
That works. Just make sure you have area sufficient for a pile. A lot of folks here have it dumped next to the street in front of their homes and then wheelbarrow it to wherever they want it on their lot. A bit of a hassle, but not too heavy or hard to move as long as you have a good bedding fork.
How close could a truck get? Is there somewhere else on your lot that chips could be dumped? It'll be work to have to wheelbarrow it in but shouldn't be too difficult.
Conversely some cities have chips that residents can pick up for free, if you have a pick up that is small enough to fit to...
I'd cover the whole thing, working on the muddy parts first. If you leave the rest uncovered it can possibly end up muddy as the rest, so why let it get to that point?
See if a tree company will dump off a load or two of chips for you, a single good pile can cover quite a lot of area (or be...
In western WA I'd go with deep litter and not sand/gravel. Think something like the forest floor. Keeping it dry is not the goal, having good drainage is. Your trees are potentially already one source of materials. Depending on what you have on your lot, you can use materials from your trees...