What you'll need: Patience, husk of the pit
I've seen this before and want to try it sometime. Basically, you start a new character. In the first mission, picking up the secondary weapon is optional (so if you're don't open the chest, your secondary slot stays empty! Heavy is also optional, but that comes later). Once you hit the tower, fill up your inventory with engrams (probably from another character), so that you can't pick up any more weapons. You run through the game as normal with only the primary weapon (you can change the primary, it doesn't matter). Once you hit lvl. 16, equip the husk of the pit. Make sure you have no equipable weapons at all, and the upgrade the husk to the eidolin ally. Since eidolin ally requires lvl. 20, it will be automatically unequipped(since you're lvl. 16), leaving you with nothing in any of your weapon slots. Place it in your vault, and enjoy your no weapons character!
There were several videos posted on this if you want visual stuff.