A supplier is threatening to sue over a contract we never signed

Yes, a contract can be formed without a signed written document. Under the Contract and Commercial Law Act 2017 and common law, a binding contract requires offer, acceptance, consideration and an intention to create legal relations, and it can be formed by emails, conduct or a verbal agreement. Some contracts must be in writing, for example those for the sale of land, but a supply agreement usually does not. The question is what was actually agreed: did the emails set out certain and complete terms, and did your manager have authority to bind the company? Collect all communications, prepare a timeline and get advice quickly. If a contract exists, you may still have defences, and the supplier must take reasonable steps to reduce its losses.