Yes, a barbarian can use reckless attack to get sneak attack every turn if they want
Once per turn, you can deal an extra 1d6 damage to one creature you hit with an attack if you have advantage on the attack roll. The attack must use a finesse or ranged weapon.
Whenever you have advantage, you can apply sneak attack as long as you are attacking with a finesse or ranged weapon. The source of the advantage does not matter at all.
... you can decide to attack recklessly. Doing so gives you advantage on melee weapon attack rolls using Strength during this turn.
If you use reckless attack using strength with a finesse melee weapon, you can apply sneak attack to one attack every turn. However, this does give opponents advantage to hit you until your turn comes again.