The mechanics of the feature are described within the feature, you can attack using whatever you like
The Armorer Artificer's 5th level Extra Attack feature states, in full:
You can attack twice, rather than once, whenever you take the Attack action on your turn.
If there were any restrictions in terms of what weapons you could use they would be included. An example of a feature that does include such restrictions is the Thirsting Blade Eldritch Invocation which states (emphasis mine):
You can attack with your pact weapon twice, instead of once, whenever you take the Attack action on your turn.
The mechanics of the Armorer's Extra Attack feature are contained within the feature itself, there aren't any hidden rules or restrictions. Well... except one from multiclassing:
If you gain the Extra Attack class feature from more than one class, the features don't add together. You can't make more than two attacks with this feature unless it says you do (as the fighter's version of Extra Attack does). Similarly, the warlock's eldritch invocation Thirsting Blade doesn't give you additional attacks if you also have Extra Attack.