In the case of a new vendor, there is not an actual legal requirement that you obtain a W-9, and no law that you can use to compel a vendor to give you one.
What you must do is obtain their tax ID and their tax classification, along with their address. Because they have provided you with tax ID, and have stated that they are not exempt, you can pay them without doing back up withholding, but you do need to issue a 1099-MISC on payments made to them. It is best practice to get the W-9, but if you have the TIN, you can pay the vendor.
(If you have received a CP-2100 from the IRS, you would be required to obtain a W-9; in that case, if they do not provide it, you must begin backup withholding. But that happens only after you’ve filed a 1099 on the vendor at least once.)
However, you should document your efforts to obtain a W-9 and how you obtained the TIN (even if you just write up the phone calls, with dates and what was said), and put it in the vendor file. You also want to be sure to verify the vendor is legitimate; PO boxes are a signal to look more closely. For example, check the website of the Secretary of State of the vendor’s home state.
For more on vendor validation, see Vendor Due Diligence: Verifying New Vendor Legitimacy on TAPN.