I have the same issue with the camp chest not responding when clicking on it, though I'm not on multiplayer.
To access the things in your chest, while at camp, click on the "Camp Inventories" icon (bottom left of the screen, blue circled icon that looks like a chest). Then click on the "Traveller's Chest" on the left, it should be there in the same menu as your party members' portraits/items.
From there, you can access the items in your chest.