Although Mass Effect Legendary Edition already has modifications such as FOV correction, it is still not as modifiable as the original games due to the absence of a set of tools such as ME3Explorer. This will come with time and a better understanding of how they have changed their .pcc files, a proprietary version of the Unreal Engine package files. One step on the way to making the legendary edition more adjustable is that the the game console has been enabled, thanks to Mgamerz.
Earlier modifications to recover the console involved renaming the .dll files and running the risk of being wrapped by the DRM of the games. The latest modifier appears to be the first to play directly with .pcc files, providing a spare Engine.pcc for all three games. If used to replace the existing file in the appropriate BioGame / CookedPCConsole directory (be sure to back up the original first), open the full console by pressing ~ and a mini-console by pressing the Tab key. (If you use the tab to get rid of the UI in photo mode, note that scrolling to the left does the same.)
You can find lists of console commands for each game on the page Mass Effect wiki. While some of the advanced ones, such as those that allow you to add specific items, may conflict with Legendary Edition modifications, all 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 want.
Other changes to the Legendary Edition allow you to activate more same-sex romances (though only for the default Shepards at this time), add more suits for ME2 and ME3, and run with infinite sprint and Mako momentum, but it will be exciting to see how the mod scene grows to the point where it was possible in the original games.