Wood:
Atlanta Hardwoods went out of business. Sad news, they were a great source.
Good prices on BB ply and MDF at Rugby Architectural Building Supplies. They have multiple thicknesses. Call ahead for availability, they have a will-call window from what I understand.
Specialty woods and veneers in midtown called Carlton’s:
http://www.rarewoodsandveneers.com/
Highland woodworking is a good source:
Rockler has multiple locations in the city:
http://www.rockler.com/retail/stores/ga/sandy-springs-store
http://www.rockler.com/retail/stores/ga/kennesaw-store
(and maybe more)
Woodcraft has two locations around town:
(Woodcraft of Atlanta, and Woodcraft of West Atlanta. The others are not the same.)
Plastics:
Professional Plastics has good deals on acrylic sheet, especially if you can pick up locally:
https://www.professionalplastics.com/
Piedmont plastics, they have a scrap bin and good customer service, says @jamely :
https://www.piedmontplastics.com/locations
(Mailorder for acrylic: estreetplastics.com has the best prices.)
Leather:
Tandy leather has 2 location in Atlanta. Really good customer service.
https://www.tandyleather.com/en/
General materials:
Uline in Gwinett county:
Sam Flax art supplies in midtown has many materials:
https://www.samflax.com/
IKEA in Midtown:
Desperate times:
The following places will do in a pinch but are not recommended due to price/other considerations.
Hobby Lobby & Michael’s: Expensive for what you get, but reliable source of hobby wood and other almost ready-made things like frames and plaques.
Big Box hardware: very limited thin stock
Ace Hardware locations: Some thin stock, but unreliable supply.
Any others?