It's certainly possible it's not required to issue a 1099-NEC for this work.

Typically, not only do you not pay them the payment that puts you at or over $600, until they provide the W9, you also should have a signed lien release to avoid finding out they filed a mechanic's lien on that property.

Here's a nice explanation of the due diligence for your client to follow after-the-fact:

https://amynorthardcpa.com/contractor-refuses-w-9/

In the next round of correspondence, I would point out that a copy of these attempts will be sent to the State (if contractors are registered in your State) and to the insurance broker (assuming your client got proof of coverage for that job site). Nothing like threatening their ability to do their work elsewhere.

Also, you might be able to find their EIN from the previous referrals or even the Secretary of State's office.