We know the original Widow isn't human either, as the ME2 description states they can't fire it without shattering their shoulder.
I'm going to extrapolate from Grunt's logic on this. In one of his first ME2 converstations, he is talking about killing stuff (typically) and mentions the relative squishiness of different species. Apparently, Humans, Asari, and Salarians are all "soft", while Quarians and Turians are tougher.
Considering this, it is likely that the former 3 races cannot fire the weapon without problems if humans can't. The idea of Krogan inventing a precision weapon is just absurd. That leaves us with Quarians and Turians as the only realistic origins. (unless volus are secretly super tough)
Also considering that, in the Geth propaganda video, Shepard and Legion specifically refer to the weapon as the same one Legion used before, it is likely that the Widow was the actual weapon used by the geth platform in question, as if it was just Shepard drawing from known memories, why wouldn't he have used a more common sniper rifle such as the Mantis or Viper? (Widow isn't even in your inventory by that point in the game and any non Infiltrator/ Soldier shep would never have used it at all). It was found on Rannoch during a time period when quarians were only a minor council species, and considering quarian technological innovation abilites, it is unlikely that they would need the turians to develop a weapon for them.
Quarian still seems like the most feasible origin for the widow.