I'd probably check with some of the companies that make and sell oilcoth dusters as to how to clean. I know they don't recommend washing because it removes the was coating. But if the choice is throw it away or wash it, I'd gamble on the wash and recondition.

there are conditioning kits that you can buy to re-condition the fabric.

From a Sailor's ad.

"A benefit of wax cotton or oilcloth gear is that reproofing will restore the rugged waterproof qualities even after years of use. "

Do a google search for Sailor's re-waterproofing or re-conditioning kets.

Hang the coat on a sturdy hanger outside and hose it down really well.  You can use some diluted laundry detergent and a brush to clean off any ground in dirt on the outside (oilcloth).  Once the outside is clean and dry, turn the coat inside out and do the same thing.  Make a dilute laundry detergent solution and a softer brush and scrub the inside and lining of the coat. Once the inside is totally clean and dry, turn it back right side out and if needed, you can buy a product from the manufacturer to re-oil the outside.

Yes, they are heavy and smelly, but you can't beat one on a cold wet day if you're stuck outside.