Yes, chickens can eat wet food. In fact, when they're sick and off their feed, that's how you get them to eat it is by mixing it with water to make a mash. However, you can NOT feed them food that is in any way moldy or starting to get moldy. Mold starts growing before you can see it. They get very sick from this.
Feeders kept outside where they can get rained/snowed on is one of the points that leads to the debate on where best to keep their feeders: inside the coop or outside. It comes down to what works best for your set up, preferences and situation. For me, I keep food inside my coop and food outside but under a covered, always dry area of my barn. I keep water in those 2 same places as well. I would never be able to keep food outside not under cover because it would be constantly getting rained on and wasted.
Hope this helps,
Guppy