ECO supplies you with work vest and hard hat. They’ll have safety glasses and work gloves on the boat. The only PPE you need is steel-toes.
As others mentioned, every boat will have washer and dryer as well as sheets, blankets, pillows, towels. A lot of guys bring their own bedding, towels, etc., but I recommend you just get by on what is provided for the first hitch or two until you see what’s available and nail down a permanent berth. Sometimes Chouest will move you around on short notice so you want to pack light. I’ve never used rain gear in 3+ years.
As “training captain” you will spend most of your time on the bridge. Your time on deck will be primarily for orientation and inspection of safety gear but probably no dirty deck work.
PM me if you want more details.