Can You Download Animal Crossing: City Folk on Nintendo DSi?

No. Animal Crossing: City Folk is a Wii game (disc-based). The DSi Shop only distributed DSiWare (native DSi apps) and a few DSi-enhanced DS titles—never full Wii games. Hardware architecture (PowerPC vs. ARM) and storage media (DVD vs. cartridge/SD) are incompatible.

To play City Folk, you need a Wii or Wii U (backward compatible) and the game disc.