My son is keen to use a lizard folk with bite attack as a character we are assuming the second attack can be a bite as the offhand attack which seems reasonable but then he also wants to use a shield the argument being that he has a free hand.

Nope

I assume by "dual wielding" you mean the Two-Weapon Fighting mechanic. You can find the rules for this mechanic in the free basic rules on DnD Beyond here. It requires your character to be wielding two light weapons, one in one hand and one in another. Bites cannot be used with this mechanic since it's not light nor wielded in a hand.

There is a feat, Dual wielder, that allows one to use Two Weapon Fighting with melee weapons that aren't light. However, they still need to be wielded in hand.

If your character has access to Extra attack from their class, they can use their natural attacks as a part of their Attack action as normal, just not with Two Weapon Fighting.

No, it cannot.

Two-Weapon Fighting (PHB, p. 195) works as follows (emphasis mine):

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. [...]

As far as I understand, your son wants to attack with a weapon in his main hand, and then attack with a bite as a bonus action, all while wielding a shield in his off hand.

This combination is, unfortunately, not possible, since a bite is an unarmed attack, and therefore not a light melee weapon. Hence, it doesn't qualify for two-weapon fighting.

Alternative solution: monk

You can, however, accomplish something similar using the Monk class, and more specifically, the Way of the Kensei Monastic Tradition found in Xanathar's Guide to Everything (page 34).

This way, you can use your action to make an attack (or multiple, once you have extra attack) with a monk weapon of your choice*, and then use your bonus action to make an unarmed strike - which is a feature of the monk, and not otherwise possible.
_{*which can be "shortswords and any simple melee weapons that don’t have the two-handed or heavy property" (PHB, p. 78), as well as "any simple or martial weapon that lacks the heavy and special properties" (Kensei weapons, XGtE p. 34).}

Unfortunately, this still doesn't allow you to use a shield, because the monk's abilities don't work while wielding a shield. For this reason, I propose using the Way of the Kensei, which provides the following ability, which is mechanically similar to using a shield:

Agile Parry. If you make an unarmed strike as part of the Attack action on your turn and are holding a kensei weapon, you can use it to defend yourself if it is a melee
weapon. You gain a +2 bonus to AC until the start of your next turn, while the weapon is in your hand and you aren't incapacitated.

Unfortunately, this still requires you to make an Unarmed Strike as part of your Attack action. However, once you're level 5, you can make two attacks as part of an Attack action, allowing for the following setup:

1. Attack action attack #1: attack with a kensei melee weapon or make an unarmed strike (e.g. a bite). You still have to be wielding a kensei melee weapon in one of your hands.


2. Attack action attack #2: attack with an unarmed strike (e.g. a bite). You now have +2 to AC until the start of your next turn (see Agile Parry feature), which is equivalent to using a nonmagical shield.


3. Bonus action attack: attack with an unarmed strike (e.g. a bite)