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 beets, cover, and steam until tender, 6 to 8 minutes. Chill beets for at least 1 hour. Chill artichoke hearts at the same time.
Mix vegan mayonnaise, mustard, vinegar, and salt together. Combine beets, artichokes, celery, and macadamia nuts in a bowl. Add dressing and toss; serve chilled.
