There is no official equivalent in 5e.
However, a barbarian who doesn't own anything is can still be a viable combatant. As barbarians can add their Con bonus to their AC when not wearing armor and also halve most of the damage they take, you really don't need the AC bonus that the Vow of Poverty feat provides. Remember that in 5e, everyone's AC is lower than the equivalent creature in 3.5e. Some races, such as the lizardfolk or tortle, provide an alternative unarmored AC that could help, especially in the early levels.
In order to deal damage on par with a barbarian who does own things, you could either take the Tavern Brawler feat or choose a race with natural weapons (lizardfolk, perhaps?) to increase your unarmed damage. Alternatively, or additionally, you could take the Martial Adept feat and choose disarming strike in order to steal the weapons from opponents. Borrowing is not the same as owning, after all.
You will run into trouble by never owning a magic weapon to use against creatures immune or resistant to non-magical attacks. This can be dealt with by any spellcaster in the party casting Magic Weapon or Elemental Weapon on an improvised or stolen weapon you pick up.