[RPG] Can an Armorer artificer using the Guardian armor model wield a shield or weapon in one Thunder Gauntlet, and still attack with the other gauntlet

Update regarding the Booming Blade errata (6/9/21)

The Booming Blade errata adds one change relevant to this answer: the weapon used as part of the melee attack must be worth at least 1 sp.

Regarding Thunder Gauntlets, the weapon for Booming Blade would be "your armored fist". So the question (for your DM, probably) is, how much is your armored fist worth?

If you verify your gauntlets/gloves are worth at least a silver, the original answer is still good.

Original answer below:

The description of the Booming Blade cantrip says (SCAG, p. 142; emphasis mine):

As part of the action used to cast this spell, you must make a melee attack with a weapon against one creature within the spell's range, otherwise the spell fails. On a hit, the target suffers the attack's normal effects, and it becomes sheathed in booming energy until the start of your next turn.

Yes, you can use Thunder Gauntlets as part of Booming Blade

This is because Booming Blade requires a melee attack with a weapon, and Thunder Gauntlets states:

Your armored fists each count as a simple melee weapon

Yes, you would get the debuff effect from Thunder Gauntlets

The spell effect states that the target takes the normal effects of the attack, which includes both the damage and the distracting pulse.

Your armor's gauntlets don't occupy your hands while worn

While in use, armor is worn, not carried. So the gauntlets that are part of the armor you are wearing don't occupy your hands, even while they "count as simple melee weapons". This means that even though you have a weapon available on each hand, your hands are still free to perform the somatic components of spells or perform any other action. (This versatility is one of the benefits of the Arcane Armor feature.)

You can use your arcane armor as a focus while wearing it

Normally, you need to (be able to) hold the material component of a spell in your hand in order to cast it:

A spellcaster must have a hand free to access a spell's material components -- or to hold a spellcasting focus -- but it can be the same hand that he or she uses to perform somatic components.

So you might be worried that you have to take off your armor and carry it in order to cast spells with it. However, the Arcane Armor feature specifically overrides this general rule, because the ability to use it as a spellcasting focus appears in the list of benefits you gain while wearing the armor, not carrying it:

You gain the following benefits while wearing this armor:

• [...]
• You can use the arcane armor as a spellcasting focus for your artificer spells.
• [...]

In fact, since you gain these benefits only while wearing the armor, this means that you actually cannot use your armor as a spellcasting focus while holding it. Contrast this with infusing a suit of armor with Enhanced Defense, which would technically allow the artificer to use the armor as a focus while holding it rather than wearing it. (Of course, if you infuse a suit of armor with Enhanced Defense and then also use your action to turn it into your arcane armor, then you would be able to use the armor as a spellcasting focus while either holding or wearing it.)