Yes, you can make 1 greataxe attack and 2 claw attacks
There are 2 parts to this question:
- Can you wield a weapon with clawed hands?
- Can you use your claws while wielding a two-handed weapon?
Fortunately, the answer to both of these is yes.
Having claws doesn't prevent wielding weapons
The main concern here is the first clause of the Form of the Beast's "Claws" option: "Your hands transform into claws," and whether this implicitly prevents you from doing anything that would normally require hands, such as wielding weapons or manipulating objects. However, there are a number of clawed creatures that can also wield weapons, such as a mezzoloth:
Multiattack. The mezzoloth makes two attacks: one with its claws and one with its trident.
As well as the dragonborn player race, which has no racial limitations on weapon use despite having clawed hands:
Their hands and feet are strong, talonlike claws with three fingers and a thumb on each hand.
Hence, there is no implicit mutual exclusion between clawed hands and weapon use. So, given that the Form of the Beast feature doesn't specify that your hands become unusable as normal hands while transformed, there is nothing preventing you from wielding weapons while your hands are transformed into claws.
You can attack with a two-handed weapon and then your claws
You might also think that a two-handed weapon would render your claws unusable, unless you attack with the weapon and then drop it. However, post-erratum, the two-handed weapon property says (emphasis added):
This weapon requires two hands when you attack with it.
This means you can attack once with your greataxe in both hands and then let go with one hand in order to make a claw attack, without dropping the weapon. This satisfies the requirements of the Form of the Beast feature in order to gain the additional attack: you have taken the Attack action on your turn and made an attack with your claws. Hence, you can make one attack with a weapon (even a two-handed one) and then 2 attacks with your claws in a single attack action.
You can even use two-weapon fighting and your claws together
With some creative object interaction, you can even make 4 attacks per turn: 2 attacks from your claws and 2 weapon attacks using two-weapon fighting. For instance, you can begin your turn wielding 2 shortswords. You take the attack action and make one attack with a sword, then use your bonus action to attack with the other sword. Then, you put away one sword and use the 2nd attack of your action to attack with your claws, allowing you to make one more attack with your claws, for a total of 2 shortsword attacks and 2 claw attacks. On your next turn, you start with 1 shortsword in hand and one free hand. You make your two claw attacks first, then you pull out your 2nd shortsword and make your 2 shortsword attacks using two-weapon fighting.
Note that, per the two-weapon fighting rules, you don't add your ability modifier to the 2nd shortsword attack, so I don't think this will improve your raw damage output over a greataxe. However, this might be useful for situations such as fighting a mage, where more attacks means more chances to break the mage's concentration.