Learn Italian: Devi inviare la dichiarazione entro domani. - You must send the declaration by tomorrow.

Entro translates to "by" or "no later than." It is used for deadlines when an action must be completed before a certain time. Fino a means "until" and is used when an action continuously happens up to a certain point (e.g., "I will wait until tomorrow"). Since sending a document is a single action, entro is correct here.

Could I say prima di domani instead of entro domani?

You could, but the meaning shifts slightly. Prima di domani strictly means "before tomorrow" (meaning it must be done today). Entro domani means "by tomorrow," which usually includes tomorrow itself as the final acceptable day to send it.

What kind of "declaration" (la dichiarazione) is this sentence likely talking about?

In Italian adult life, when you hear about a strict deadline for la dichiarazione, it almost always refers to la dichiarazione dei redditi (the tax return). It can also refer to other formal statements or customs declarations.

Why is it devi inviare rather than hai bisogno di inviare for "you must send"?