Other than local suppliers like chem shops, hardware stores and pharmacies, there are many places online that sell pyro supplies. Many require a copy of your ID for age verification, but not all.
If you have ordered from any of these places (or other places I didn't mention) and would like to tell how it went (did you get what you were expecting, how long did it take etc) email me and maybe I'll put it up here.
Skylighter - Probably the biggest and best online supplier. They have a massive product selection and good prices. They have many books and videos on pyrotechnics, as well as high quality pyro tools. You must be on file with them to order, which means sending a copy of your drivers license or other ID.
Firefox - Similar selection to Skylighter. They have some products Skylighter does not and vice versa. You must be on file with them to order.
Pyrotek - Pyrotek sells a wide variety of pyro, rocketry and chemistry supplies. They have a large selection and decent prices. Warning! I have heard some bad things about this place. For example, I got an email from somebody saying they ordered fuse here, never got it, and did not get their money back. I have also heard from numerous people who report having no problems at all. I have ordered from them with no problems.
Dawntreader Pyrotechnics - Haven't heard much about them, but they have quite a few chemicals and decent prices.
Wolter Pyro Tools - Nice tools for rockets, comets etc.
Pyrosupplies.com - "High quality and hard to find pyrotechnic supplies"
Precocious Pyrotechnics - Non-chemical supplies like mortar tubes and other cardboard products.
LORTONE, inc. - Rock tumblers often used as ball mills. Lists local distributors.
United Nuclear - No ID required, they have a lot of good products, but prices are very high for many things. Shop around before buying here. The no longer carry things like KClO4 and dark flake Al because too many losers ordered them and got in trouble.
Stanford Systems Aerospace - A rocketry supplier. Warning! Many people (including myself) have ordered from here and had serious delays or have not received orders. DO NOT ORDER FROM HERE!
eBay - You can sometimes find chemicals like kno3, sulfur, and potassium perchlorate here, but prices will most likely not be very good.
Cannonfuse.com - They sell fuse and a couple sizes of tubes. I have ordered from here with quick service, the price for fuse is far better than United Nuclear. Others have reported poor service, but most people I have heard from have not had problems.
Discount Pyro - Small selection, but very cheap. Requires ID. I have ordered here with no problems.
Pyro Plastics - Plastic aerial shell casings, class B shells listed and a mention of expanding to Class C sales.
Pyrohobby - A new supplier, sells a few chemicals and does not require ID.
Pyrostuff - Another new supplier, only has a few chems and tubes.
New!Pyroworks - Never heard of them until somebody emailed me a link, if anyone has experience with them, let me know how it went.