Learn Italian: Senza internet non posso lavorare. - Without the internet I cannot work.

In Italian, to make a sentence negative, you simply put non immediately before the conjugated verb. So "I can" is posso, and "I cannot" becomes non posso.

Why is the verb lavorare left in its dictionary form?

Posso comes from the verb potere (can / to be able to). When you use verbs like can, want, or must, the action verb that follows stays in its infinitive form (ending in -are, -ere, or -ire). Since posso already tells us who is doing the action (I), lavorare (to work) doesn't need to change.

Do I need to use "the" before internet, like "senza l'internet"?

No. Even though it translates as "without the internet", Italians often use the English loanword internet without an article after certain prepositions like senza (without) or su (on). So senza internet is completely natural.

How would I change this sentence to say "Without the internet we cannot work"?

You only need to change the verb posso (I can) to its "we" form, which is possiamo. The rest of the sentence stays exactly the same: Senza internet non possiamo lavorare.