Learn Italian: Possiamo pagare con la carta?

Because it follows the modal verb possiamo (we can). In Italian, when you use a modal verb, the second verb always stays in its infinitive (base) form. This is similar to English, where we say "we can pay" instead of "we can we pay".

Where is the word for "we" in this sentence?

The "we" is built right into the verb. The ending of possiamo tells us that the subject is "we" (noi). In Italian, you usually drop the subject pronoun unless you want to add special emphasis.

Does la carta just mean any card, or specifically a credit card?

In the context of shopping or restaurants, la carta is the everyday shortcut for saying credit card (la carta di credito) or debit card.

I learned that prepositions combine with the word for "the". Why are con and la written separately?

While prepositions like a, di, and in always merge with definite articles (creating words like al or nella), the preposition con (with) is an exception. In modern Italian, it usually stays separate from the article.