Otdfbt

Lizardfolk have a feature that gives them natural weapons in the form of their jaws. This bite deals 1d6+Str damage.

Hungry Jaws. In battle, you can throw yourself into a vicious feeding frenzy. As a bonus action, you can make a special attack with your bite. If the attack hits, it deals its normal damage, and you gain temporary hit points (minimum of 1) equal to your Constitution modifier, and you can't use this trait again until you finish a short or long rest. (VGM 113)

Assuming you transform into a beast shape with the ability to perform a bite, the Hungry Jaws trait should be available to you, as racial features transfer over when you transform if the new form is capable of them.

Given this circumstance, would you use the Lizardfolk's racial bite, or the bite attack belonging to the new form?

You get to choose

The rule on Wild Shape states

• 
  You retain the benefit of any features from your class, race, or other source and can use them if the new form is physically capable of doing so.

What this suggests is that you do not replace the racial version of the bite attack. You have two bite attack options: one granted by your race and one by your Wild Shape.

You can choose which one to use (although both would key off the beast's strength mod, so you're really only choosing which base damage die to use) because the beast you have Wild Shaped into also has the ability to make a bite attack and that's all Hungry Jaws needs.

Reading other answers, it's clear that there is the possibility a DM might interpret the Lizardfolk's bite attack to be somehow unique in nature and not usable by a beast form with a bite attack.

I see no indication of this personally, but the if the DM rules that this is the case, then the Hungry Jaws racial ability would be unusable since this interpretation requires the unusable Lizardfolk bite ability.

Hungry Jaws should be available

First, let see how the traits are defined :

Wild Shape: You retain the benefit of any features from your class, race, or other source and can use them if the new form is physically capable of doing

Hungry Jaws: In battle, you can throw yourself into a vicious feeding frenzy. As a bonus action, you can make a special attack with your bite [...].

(emphasis mine)

Hungry Jaws is described as a frenzy, a mental state, the PC goes into. It doesn't sounds like it rely on their physical form so a Wild Shaped lizardfolk should be able to use it.

The Lizardfolk bite attack though...

For the bite attack, as a DM, I would rule that it's clearly linked to the Lizardfolk morphology and the PC wouldn't be able to use it when transformed. Even if the new form is physically able to bite something, it's not physically able to so as a lizardfolk. Physically, a mouse is able to bite something but it shouldn't be the same as a lizardfolk maw.

The PC could still use the Hungry Jaws trait if the new form have a bite attack.

Hungry Jaws is not available

When using Wild Shape, you can only use features if "the new form is physically capable of doing so." Your new form physically doesn't have lizardfolk jaws, so the features available for lizardfolk biting do not apply.

On the other hand, it won't break the game since Hungry Jaws is a "once per rest" feature.

It might break the game if you allow it for all features, however...

Wild Shaped kobolds using darkvision, or Wild Shaped bugbears taking advantage of long-limbed, or Wild Shaped tieflings having wings.