The two words are adverbs, but their root words have completely different uses. Vagariously comes from a noun, "vagary." Vicariously comes from an adjective, "vicarious." The addition of "-ly" makes the words adverbs, in that their roots are now used to describe the qualities of a literary action, or verb.
The root words of these two adverbs significantly differ. Merriam-Webster (merriam-webster.com) says vagary is "an erratic, unpredictable, or extravagant manifestation, action, or notion." In essence, a vagary is an unexpected or wholly unusual deviation from an expected pattern. An action performed vagariously performs in contrast to what is expected or performs inexplicably.
Vicarious has many subtle definitions, but is an adjective that describes an item "serving instead of someone or something else," according to Merriam-Webster. Vicarious is often used in its adverb form; vicariously, which means the subject in a sentence experiences an action secondhand.
Vagary seems to be a slightly older word than vicarious, but both derive from latin words. Vicarious can be traced back to "vicis," which mean "change" or "stead." Vagary, is suspected to have come from the Latin "vagari," meaning "to wander." Merriam-Webster says the first known use of "vagary," 1579, predates the first known use of "vicarious" in 1637; however, this difference in age is somewhat insignificant in light of the long history of the English languages, which can be approximated to the fifth century A.D.
Vagariously can be used when someone needs to describe an action as inexplicably deviant from the norm. "He walked vagariously across the field," or "The weather patterns changed even more vagariously than usual."
Vicarious is best used when describing the extent to which the direct object of a sentence is experiencing something as a result of the actions of an indirect subject. "The father relived his youth vicariously through his son's college antics," or "The public vicariously experiences the war through the images and words of the news media."