I've never quite understood what each Mozilla does, but AIUI, the Firefox development is all done by Mozilla Corp and the nonprofit does stuff like make those cute videos about how Firefox is going to save the world and make us all smiley and multiethnic.
I've talked to a number of Mozilla employees, and they also seem confused about the relationship between the corp and the foundation.
The only way that the Corporation could do shenanigans, is by paying their employees higher wages than would be necessary to hold them or is considered reasonable for the job that they do.
It was set up, because there's limits to how much money a non-profit is allowed to put to the side, which would have limited Mozilla's strategic flexibility.
I don't think that actually is correct. The Mozilla Foundation is a non-profit, as has already been pointed out above. As such, they specify a mission and then are legally forced to invest all money that they make/get into this mission.
And in their mission statement, they specify that "Individuals’ security and privacy on the Internet are fundamental and must not be treated as optional." (Principle 4 in the Manifesto).
So, they cannot legally do shady stuff. And even if that means nothing else to you, the rule with law violations still is "Innocent until proven otherwise".