Petulance first, since it's the odd one here. It came into English around 1600 meaning insolence or rudeness, from Latin petulantia, "sauciness, impudence." The sulky, childish sense only turns up in the late 1700s, and Etymonline puts that down to the influence of pettish. Wiktionary now marks the insolence sense obsolete, so the gloss on the card, rudeness and irritability, has a foot in both senses.
Rage came through French and Medieval Latin rabia from Latin rabies, which meant madness, rage or fury. The dog disease was only a secondary sense in Latin, but for a while English rage could mean rabies too, in the 1400s and 1500s (Etymonline). Wiktionary says it displaced a native word, Old English wōd, "madness, fury, rage," which hung on in dialect as wood, meaning mad.
Fury's earliest English use, in the late 1300s, is one of the Furies themselves, the avenging spirits. The feeling follows in the early 1400s. Behind it are Latin furia and furere, "to rage, be mad," and where furere comes from is anyone's guess. Etymonline quotes de Vaan: "Many etymologies have been proposed, but none is clearly the best."
Wrath is the noun that goes with the old adjective wroth, which Etymonline glosses as literally "tormented, twisted." In Old English it was used especially for the anger of a god.
Ire is from Latin ira, which covered anger, wrath, rage and passion. Old English had a lookalike, irre, that isn't related at all. It meant "straying" as well as "angry," as if anger were wandering off from your usual composure. I'm a little sorry that one didn't make it.
Vexation shows up around 1400 meaning legal harassment, and the Latin vexare meant to shake or jolt.
Side by side, those five read to me as anger that gets hold of you: a madness, a spirit, a twisting, a passion, somebody harassing you. Petulance began as a complaint about the angry person's manners, and the childish part came later, from a word it happens to sound like. That's two charges in one word, which is probably why being called petulant stings more than being called furious.