The Crimson Dye merchant is on the north side of the map, SW from Fort Sakai. You unlock his store after defeating him in the Hidden Cove Tournament at Fune's Refuge (advance the story if you don't have the tournament available)
The Horse Dye merchant is on the east side of the map, directly west of Thieves Rest Lighthouse and Thieves Rest Village. You can purchase a dye once you have collected enough Sakai Banners.