An all expense-paid trip to Florida will be given to (whoever/whomever) wins the sales contest.
This would be whomever?
And...Please direct my inquiry to (whoever/whomever) is in charge of quality control.
this would be whoever?
The trip to Florida is given to whoever, because whoever is the subject of the verb "wins." The noun clause "whoever wins the sales contest" is the object of the preposition, "to."
The same explanation applies to the second sentence.