My son is keen to use a lizardfolk 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.
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:
- 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.
- 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.
- Bonus action attack: attack with an unarmed strike (e.g. a bite)
According to RAW: No.
According to Dad GM: Yes.
The rules as written clearly differentiate between a light melee weapon and and an unarmed strike. While an unarmed strike is considered a weapon, it is a simple melee weapon with no other properties (see Weapons chart on PHB 149). Light is a weapon property that unarmed strikes do not have.
Two-Weapon Fighting states (PHB 195, 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.
Unfortunately this excludes unarmed strikes from two-weapon fighting.
Work-arounds: By RAW 5th level fighters, barbarians, paladins, rangers and monks gain an extra attack. Your son's character could even do a two-weapon attack with each hand on their first attack + bonus attack and then a bite on their second attack. At 11th and 20th level a fighter gets a 2nd and 3rd extra attack.
Deviating from RAW, I see no reason why the two-weapon fighting rules could not be adjusted to replace a one handed light melee weapon attack with the lizardfolk bite. This would not break the game and would add a cool-factor that would make the game a lot more fun for your kid. You could even add silver teeth caps or something similar and call his teeth light-melee weapons. This would leave the fact that they are not being held in hand as the only discrepancy with the rules. I think it would be a safe stretch to consider the teeth in the mouth of a lizardfolk as the equivalent of a light weapon being held in hand, and that a bite attack could replace the second light melee weapon attack.
Additionally I would expand this exception to apply to the warlock's pact weapon (PHB 107) so that if your son's lizardfolk character is a warlock they could create a pact weapon on their teeth and attack twice at 5th level using the Eldritch Invocation Thirsting Blade (PHB 111).
Can the character attack with two weapons, one being a bite, while also wielding a shield? The rules on two weapon fighting don't mention shields because it is assumed both hands are holding weapons and thus a shield cannot be used. Allowing the shield adds on to existing rules, rather than replacing one equivalent action with another (like the bite for the weapon attack). This gives the character additional features that no other character has and could give them an unfair advantage: two attacks + shield AC bonus. This is the challenge of bending the rules. Often changing something slightly opens up more possibilities for bigger changes. Personally I would not allow, or would greatly discourage this additional feature. While it doesn't make the player OP it does give them an edge over all the other players.
Teaching kids about appropriately bending the rules: As a Dad GM myself I often run into these situations with my kids and their friends. The challenge I find is to balance rule-following with knowing when rules can be bent and how to do that responsibly. In a a situation like this I would be upfront with the player and tell them look, this isn't allowed by the rules, but with a little tweaking we can make it happen. I don't think it will give your character an unfair advantage, and it seems really cool. What do you think? Should we let your character do this? I would also explain to them the details of how we would change the rules and the details of the shield use and let them be a part of the decision, fully understanding how it will impact their character and other players. I find it important to let them have a hand in the decision and also in the process of thinking through why some rules are ok to change and some might not be.