Does Blizzard allow people to buy and download the first diablo game online? Or do I need to purchase a physical copy?
No, you cannot purchase the original Diablo game, in digital or physical form, from Blizzard's online store. The game still appears to be available on Amazon for either the PC or PlayStation.
If not, you can try searching for other releases or bundle. In 1998 there was a re-release bundle called Diablo + Hellfire. Blizzard also released a Game of the Year Collection in 1998 that contains the original Diablo. The Blizzard Anthology, released in 2000, also contains the original Diablo game. The Diablo Gift Pack contained the first two Diablo games but no expansions. In 2001, a Diablo: Battle Chest was available which contained the original Diablo, Diablo II, and the Lord of Destruction expansion.
Diablo II, however, is still available from Blizzard's online store. The original Diablo II game, Lord of Destruction expansion, and the Diablo II Battle Chest (which includes the Diablo II, its expansion, and strategy guides for each) are all there. However, the Battle Chest for Diablo II does not include the original Diablo game, but Diablo II and its expansion are available for digital download.
Diablo 1 is now treated as abandonware and can be downloaded here. I recommend also getting the 1.09 patch and updating if you're interested in playing this game for any length of time, which can be found here
You can utilize GameRanger or Hamachi to play multiplayer games on simulated local networks nowadays, despite Battle.net discontinuing support for the game.