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Artificer attacks follow the same rules as all other classes
All PCs get a single attack starting at level 1. Some classes naturally give the character 2 or more attacks as they level. Nothing in the description says that just because the player has two fists they can get two attacks when they start out.
Thunder Gauntlets. Each of the armor's gauntlets counts as a simple melee weapon while you aren't holding anything in it, and it deals 1d8 thunder damage on a hit. A creature hit by the gauntlet has disadvantage on attack rolls against targets other than you until the start of your next turn, as the armor magically emits a distracting pulse when the creature attacks someone else.
As it turns out, the Artificer Armorer does get a second attack at level 5 (which is pretty average).
But what about two-weapon fighting? I hear you ask:
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. You don't add your ability modifier to the damage of the bonus attack, unless that modifier is negative.
Unfortunately, gauntlets do not have the "light" property, just simple and melee. So they cannot, by default, be used for two-weapon fighting.
However, there is the Dual Wielder feat that removes that rule:
You can use two-weapon fighting even when the one-handed melee weapons you are wielding aren't light.
So if you invest in that feat, you can duke it out with both hands. Or, once you gain 5th level Artificer, you gain a second attack on your Attack Action so you can either punch twice with one hand or once with each. And if you are 5th level and have the feat you can punch three times.
It should also be noted that the gauntlets only count as a weapon, "while you aren’t holding anything in it." So if you use a shield, you only can punch with the other hand.