Yes, you can use a two-handed weapon as your pact weapon, but...
We need to dissect things a little bit to understand what exactly that means.
The problem with Hex Warrior and Pact of the Blade is that people’s intuitive assumption of what’s going on isn’t actually what the rules say. I had this confusion, my DM had this confusion, Critical Role appears to have this confusion: it’s confusing. Probably because they had to add Hex Warrior on top of the already-existing Pact of the Blade, and so had to word things awkwardly.
So, let’s go through the different features really carefully.
The influence of your patron also allows you to mystically channel your will through a particular weapon. Whenever you finish a long rest, you can touch one weapon that you are proficient with and that lacks the two-handed property. When you attack with that weapon, you can use your Charisma modifier, instead of Strength or Dexterity, for the attack and damage rolls. This benefit lasts until you finish a long rest.
(Xanathar’s Guide to Everything description of the Hex Warrior feature)
This is the ability you get when you first choose the Hexblade patron and gain the Hex Warrior feature.1 It allows you to pick some weapon that already exists, and use Charisma for attack and damage instead of Strength or Dexterity. This ability requires that the weapon you choose, that already exists, not have the two-handed property.
Now then, look at this:
If you later gain the Pact of the Blade feature, this benefit extends to every pact weapon you conjure with that feature, no matter the weapon's type.
(Xanathar’s Guide to Everything description of the Hex Warrior feature, emphasis mine)
This is the next sentence after the previous quote; it’s part of the same 1st-level Hex Warrior feature. It specifies that the whole Charisma-to-attack-and-damage thing applies to “every pact weapon you conjure with that feature.”
What is a “pact weapon?” That phrase hasn’t been used before; this is the first time we’re seeing it. We have to look to Pact of the Blade:
You can use your action to create a pact weapon in your empty hand. [...]
You can transform one magic weapon into your pact weapon [...] You can then dismiss the weapon, shunting it into an extradimensional space, and it appears whenever you create your pact weapon thereafter.
(Player’s Handbook description of Pact of the Blade, emphasis mine)
So Pact of the Blade defines “pact weapon” for us, and specifies that you can get a pact weapon in one of two ways:
Conjuring it from thin air, or
Transforming a magic weapon into your pact weapon, which you can later conjure and dismiss at will.
So here’s the two big things that people misunderstand about Hex Warrior:
The chosen Hex Warrior weapon is not a pact weapon unless you transform it into one with Pact of the Blade
If you don’t do that, you can get both a chosen Hex Warrior weapon and a pact weapon
These are two separate abilities: one to touch a weapon and make it use Charisma, the other to conjure a weapon from thin air, so it counts as a pact weapon and gets several bonuses (including using Charisma). You can touch a weapon for Hex Warrior, and then also conjure a weapon with Pact of the Blade, and both get to use Charisma. Only the pact weapon from Pact of the Blade, however, benefits from various pact-weapon invocations, like thirsting blade that gives you an extra attack with a pact weapon. In order to benefit from that feature with the weapon chosen with Hex Warrior, you would have to transform it into your pact weapon, per the second option in Pact of the Blade—but if you did, you’d be able to just choose some other weapon you have lying around for Hex Warrior after your next long rest, since it being your pact weapon means it already benefits from Hex Warrior and there is no reason to choose it again.
The weapon you touch (which has no special name) must not be two-handed. A weapon you conjure, however, can be two-handed—and it still gets to use Charisma even though it’s two-handed, because Hex Warrior says “every pact weapon [...] no matter the weapon’s type.”
Hex Warrior allows you to use Charisma for attack and damage with any one2 non-two-handed melee weapon you touch. Hex Warrior also allows you to use Charisma with any pact weapon you create with Pact of the Blade—and Pact of the Blade is not restricted to non-two-handed weapons. So if you have both, you can conjure a two-handed weapon and use Hex Warrior on it.
Along with some proficiences that aren’t relevant here.
Per long rest.