I'm playing a 3rd level Path of the Beast Barbarian. If my claws are out can a shield be used and still get 2 claw attacks in?
If one hand is free, you get two attacks
The OP asked if you can still get 2 claw attacks in.
At first it may seem one way...
Each of your hands transforms into a claw, which you can use as a weapon if it’s empty. It deals 1d6 slashing damage on a hit. Once on each of your turns when you attack with a claw using the Attack action, you can make one additional claw attack as part of the same action.
This says you can use the claw as a weapon if the claw is empty. Since it has a shield in it, the 2nd claw can't be used as a weapon.
it doesn't explicitly says that the second attack comes from the second claw, though many DMs would read it that way.
A combat round is 6 seconds long, which is plenty of time for a single claw to land 2 attacks. So you could argue that as long as one claw is empty, both attacks can be taken.
It's not overruled by more specific rules
Yes, the rules for Two-Weapon Fighting are more specific:
When you take the Attack Action and Attack with a light melee weapon that you’re holding in one hand, you can use a bonus Action to Attack with a different light melee weapon that you’re holding in the other hand. You don’t add your ability modifier to the damage of the bonus Attack, unless that modifier is negative.
But unless this is described as a two-weapon attack (it's not), this only demonstrates a precedent where if a second attack is meant to come from a second light weapon, it will say so.
The Beast Claw isn't described as two-weapon fighting or 'multiattack' in the Form of the Beast - Claws entry. In fact, it doesn't even cost a bonus action like 2-weapon fighting does. So it stands to reason that it would need to be specifically stated if the second attack was meant to have come from the second claw.
RAW, if one hand is free, you get two attacks.
Addendum: I checked the Monster Manual, and any references to Multiattack which uses specific limbs or specific attacks for their multiple attacks are always specified. Otherwise it's like, "X makes 3 tentacle attacks." The Beast Barb claw isn't even described as a Multiattack, but this is more evidence for the "it would say so" argument.