The three words in the title above, and others that share a derivation alluding to a lack of certainty or direction, are defined and discussed in this post.
The Latin adjective vagus literally means “wandering” and figuratively refers to uncertainty. The name of either of a pair of nerves that extend from the brain to the abdominal organs is taken directly from this term, and vague means “uncertain” or “lacking specificity”; the noun form is vagueness.
Some etymological sources trace vagrant, meaning “wanderer,” to early Germanic languages as a cognate with walk. However, it might also be derived from the Old French term vagant, from vagari, the Latin verb form of vagus. The word, also used as an adjective, generally refers to an itinerant person with no home or steady (or legal) employment.
A similar and related (and more colorful) term is vagabond, from the Latin gerund vagabundus, meaning “wandering.”
Vagari, meanwhile, is the source of vagary, a little-used noun meaning “aimless journey” by way of the Italian word vagare (or perhaps directly from the Latin word). The plural form, vagaries, much more common, refers to unpredictability.
Two other terms derived from vagari, one rare and the other obsolete, are the nouns divagate (literally, “wander apart”) and evagation, meaning “the act of wandering.” A more prominent derivation is extravagant, which means “excessive” or “extreme.” Interestingly, stray, meaning “wanderer” as a noun (as when referring to a stray animal) and “wander” as a verb (including in the sense of deviating from proper conduct), may be derived from extravagant, though it possibly stems from Latin by way of Old French as a cognate of street.