While Mass Effect Legendary Edition already has mods like the FOV fix, it's still not as moddable as the original games due to the absence of a toolset like ME3Explorer. That'll come with time and a better understanding of how its .pcc files—a proprietary version of Unreal Engine package files—have changed. One step along the way to making the Legendary Edition more tweakable is that the in-game console has been enabled, thanks to Mgamerz.
Previous mods to bring back the console involved renaming .dll files and were at risk of being messed with by the games' DRM. The newer mod appears to be the first tinkering directly with .pcc files, providing a replacement Engine.pcc for all three games. If used to replace the existing file in the appropriate BioGame/CookedPCConsole directory (make sure to back up the original first), you'll be open the full console by pressing ~ and a mini-console by pressing tab. (If you use tab to get rid of the UI in photo mode, note that left shift does the same thing.)
Lists of console commands for each game can be found at the Mass Effect wiki. While some of the advanced ones, like those that let you add specific items, may conflict with alterations made by the Legendary Edition all the basic commands should remain exactly the same. Try commands like "Ghost" and "Walk" to enable and disable noclip, as well as SetParagon x and SetRenegade x, where x is the number of Paragon or Renegade points you'd like.
Other mods for the Legendary Edition let you enable more same-sex romances (though only for default Shepards at the moment), add extra outfits for ME2 and ME3, and race about with infinite sprint and Mako boost, but it'll be exciting to see the mod scene grow to something approaching the scope of what was possible in the original games.