Do I need a signed contract to sue over an unpaid invoice, or will emails do?

Emails, individual invoices, and an established course of dealing between you and the client can all serve as valid evidence, and a written contract doesn't have to take the form of one single formally executed document to be enforceable. What actually matters to a court is your ability to prove the specific terms that were agreed to and that the other party accepted those terms, however that proof is assembled.

In practice, a signed proposal combined with a prior invoice from the same client that was actually paid without objection is often a stronger evidentiary combination than a lengthy formal contract that nobody involved ever carefully read or negotiated.