You cannot get a blank DS cartridge. There are manafactured for Nintendo by a company named Macronix. They use Mask ROMs, which means the game is programmed into the cartridge during the manafacturing of the cartridge, so it can't be overwritten.
They are numerous flash carts for the GBA and DS that can often hold a memory card. ROM files are put onto the memory card and can be played back on the DS.
Popular ones are R4 and Acekard.
http://en.wikipedia.org/wiki/Nintendo_DS_storage_devices
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cartridge
Route 228 (after you get the National Pokedex, insert a Pokemon Leafgreen cartridge into the Nintendo DS's GBA cartridge slot)
the cartridge
Route 208, Route 210 (After obtaining the National Pokedex, insert Sapphire cartridge into your DS).
The game ID for a Nintendo DS game can be found on the cartridge. There will be a code, which is specific for each DS game.