You need a weapon for all the smite spells (except wrathful smite)
Smite spells require a weapon
Thunderous smite says:
The first time you hit with a melee weapon attack during the spell's duration, your weapon rings with thunder ...
In fact, every smite spell has similar language specifically calling out the need for a weapon (except for wrathful smite see below). By a strict RAW reading though the weapon does not actually need to be the one that makes the attack, but you still need a weapon.
Regardless, I think it is pretty clear here that the spell is intended to utilize the weapon that you use to make the melee weapon attack even if the wording leaves a bit of a loophole. Either way, the spells do require a weapon.
The body/unarmed strikes are not weapons
The PHB says:
Instead of using a weapon to make a melee weapon attack, you can use an unarmed strike: a punch, kick, head-butt, or similar forceful blow (none of which count as weapons). (PHB 195)
Suggesting that though body parts are not weapons, the unarmed strike still counts as a melee weapon attack but explicitly not a weapon.
And the Sage Advice Compendium clarifies this and makes it explicit:
[A]n unarmed strike counts as a melee weapon attack, even though the attacker’s body isn’t considered a weapon.
Since the smite spells specifically call out a weapon in their description, that is what they require. And since unarmed strikes are not considered to be weapons, then they do not qualify.
Wrathful smite does not need a weapon
Wrathful smite is, for some reason, the only smite spell that does not specifically call out the need for a weapon.1
The first time you hit with a melee weapon attack during the spell's duration, your attack deals an extra 1d6 psychic damage.
Thus, since the unarmed strikes do count as melee weapon attacks, a character would be able to use an unarmed strike to cause the effect, no weapon required at all.
1 - Thanks @Someone_Evil for catching this.