There were art supply stores, just like now, where artists could purchase all of the necessary tools of the trade. Most larger cities would have them. An artist who lived away from the city could order via mail.

Some artists still mixed their own pigments back then (some art schools encouraged students to learn how to do this as was tradition), so an artist would need to visit a hardware store for turpentine and linseed oil, maybe canvas (if they weren’t too particular), wood for stretchers, etc. some apothecaries would carry some other supplies such as certain pigments. Most pigments were made from nature, so burnt umber would be made from burnt umber, white was usually lead, cadmium was made from a cadmium compound, etc. There were even some pigments made from ground-up mummies, insects and other things.