Yes, they can. This guide has a screenshot showing Marina giving the player a dungeness crab. Keep doing the fish errands and you'll get one eventually.
You could also try to shake up the game's RNG. Save & quit. Then reload the game to have different villagers cooking instead.
That's just Marina telling you what the secret ingredient is.
I haven't seen it happen yet so I'm not sure.
Ankha gave me a dungeness crab to complete the event too, but she wanted a fish. She then gave me hints that it was the secret ingredient like she forgot what she just gave me.Yes - Ankha gave me a dungeness crab after I gave her a turban shell. It was the last ingredient I needed to complete the event, not sure if that affects the RNG of it or not.