Place potatoes into a large pot and cover with salted water; bring to a boil. Reduce heat to medium-low and simmer until tender, about 10 minutes. Drain.
Place a steamer insert into a saucepan and fill with water to just below the bottom of the steamer. Bring water to a boil. Add carrots, cover, and steam until softened, 4 to 6 minutes.
Melt butter in a large pot over medium heat. Add onion; cook and stir until softened, about 5 minutes. Remove from heat; stir in flour until a smooth paste forms. Return to heat and pour in milk slowly, stirring constantly.
Stir potatoes, carrots, corn, salmon, green beans, peas, salt, dill, and black pepper into the pot. Cook and stir until chowder thickens, about 5 minutes. Reduce heat to low; cover and simmer until green beans are tender, about 5 minutes.