[RPG] Do natural weapons work with the Booming Blade spell

No.

Unarmed Strike

The damage from an unarmed strike is considered weapon damage for the purposes of effects that give you a bonus on weapon damage rolls.

An unarmed strike is always considered a light weapon.

Improved Unarmed Strike

Benefit: You are considered to be armed even when unarmed —that is, you do not provoke attacks or opportunity from armed opponents when you attack them while unarmed. However, you still get an attack of opportunity against any opponent who makes an unarmed attack on you.

Unarmed Strike (Monk)

A monk’s unarmed strike is treated both as a manufactured weapon and a natural weapon for the purpose of spells and effects that enhance or improve either manufactured weapons or natural weapons.

Conclusion

In regards to anyone other than a monk, an unarmed strike is a light weapon that is neither natural nor manufactured.

Exclusively for the monk (and others with monk-like abilities), an unarmed strike is a manufactured light weapon, and a natural light weapon.

Any weapon that is not intrinsic to the creature, such as a sword, a rock you pick up, a club you make.

What is a Natural Weapon?

Natural weapons are weapons that are physically a part of a creature, such as a bite, a claw, a tail slap.

Unless specifically mentioned, (such as the Monk, creature listing or racial description, etc.) no one has a natural weapon right away. For example, no human has a claw attack, a bite attack, etc. without feats, spells, or other abilities that gives that to them.

Also, an unarmed strike does not cause lethal damage, provokes an attack of opportunity, and is a very unappealing option without Improved Unarmed Strike, and/or some other way to make it better. Monk's specifically mention that their unarmed strikes count as both manufactured and natural or intents and purposes - because almost no one else's do.

If you take Skip William's advice to heart (which is not RAW) then you will find him stating, "For purposes of weapon enhancements, an unarmed strike is considered a natural weapon," and "Natural weaponry deals lethal damage without recourse to a class feature or feat, such as Improved Unarmed Strike."

Natural weapons are a kind of weapon...

There are almost no general rules on natural weapons in the PHB or Basic Rules describing what they are and how they interact with other rules. Some game features grant specific natural weapons to a character, and some monsters/NPCs have specific natural weapons listed in their statblock, but we can't derive general rules from that.

As of 2021, the Sage Advice Compendium (v2.7) contains a single official ruling about natural weapons:

Are natural weapons considered weapons?

Things designated as weapons by the rules, including natural weapons, are indeed weapons. In contrast, unarmed strikes are not weapons. They are something you do with an unarmed part of your body.

This is a pretty clear official ruling: Natural weapons are weapons. This is a logical interpretation of the rules, given that they're referred to as "weapons" in the first place.

On the topic of unarmed strikes:
As clarified in the Q&A Can natural weapons be used for unarmed strikes?, natural weapons are not used to make unarmed strikes unless a game feature explicitly states otherwise. By default, natural weapons are just another kind of weapon (separate from simple and martial weapons). However, as noted in my answer there, most features that grant natural weapons to player characters do specify that the natural weapon can be used for unarmed strikes.

...but they don't have any weapon properties unless they say they do

In an unofficial tweet from June 2015, rules designer Jeremy Crawford responded to a since-deleted tweet asking whether natural weapons have the finesse property:

Are natural weapons considered finesse weapons? A lot of them use dex for to-hit and damage, but I can't find a reference.

No. A weapon has the finesse property only if its description says so, and using Dex. to hit doesn't equal finesse.

In general, weapons don't have specified properties (e.g. finesse, light, thrown, etc.) unless their descriptions in the rules say they do. As such, natural weapons don't have any weapon properties unless they say they do.

Whether they're melee or ranged weapons is indicated by the kind of attack they can be used to make, as detailed in their description – but "melee" and "ranged" aren't weapon properties, just classifications of weapons. Beyond that, we can't assume they have any weapon properties unless explicitly stated otherwise.

Most features don't care whether you're using a weapon (or what properties it has) anyway

Natural weapons are usually used to make melee weapon attacks. (Occasionally, the description of a natural weapon specifies that it is used to make ranged weapon attacks, though I think that mostly applies to monsters that have natural weapons.)

In general, most game features that apply to or interact with an attack don't care whether you're using a weapon or not, and very few features rely on specific weapon properties. Many attack-related features trigger off "weapon attacks" or specifically "melee weapon attacks", rather than requiring a weapon or relying on the weapon's particular properties. (The rogue's Sneak Attack is one of the few notable exceptions, requiring a finesse or ranged weapon.)

As such, even if natural weapons weren't considered weapons (and regardless of whether they have any particular weapon properties), most such features would work with natural weapons anyway.