The answer is no: Mass Effect Legendary Edition is completely a single-player affair. There's no in-game co-op or any extraneous multiplayer features to be found. So if you were looking to play it with friends, you're out of luck.

Does Mass Effect Andromeda have co-op campaign?

If you're looking for couch co-op for the campaign mode, however, there is no option for playing with others during the story. That will be a solo experience. Luckily, however, the PvE multiplayer will be available from the start of the game, and you'll be able to jump in with friends at any time.

Is there multiplayer in Mass Effect?

Apparently, the answer to all of these questions were no, as multiplayer is not currently part of Mass Effect Legendary Edition.