Can a material supplier pursue payment from me, the Contractor, or the Owner?

In Florida, if the material supplier served a notice to owner/contractor, then they likely can pursue the property owner for payment. Depending on the contract, you may or may not have any liability for the actions of your subcontractor in failing to pay for the materials, but in all likelihood, the owner will look to you to make this right. 

I cannot address Virginia, but under Florida law, material suppliers generally cannot sue a general contractor such as yourself with whom they have no contract (one exception would be if you posted a payment bond on the project). A material supplier would not be able to sue the owner for payment unless the material supplier filed a timely construction lien on the owner's property and then timely sued to foreclose the lien.

Generally speaking, the Owner's contract with the GC likely has a provision requiring the GC to bond off or otherwise cause liens to be released. The GC in turn likely has similar language in its contract with you, obligating you to release liens on the property.  

If the supplier files a lien then you would conceivably have an obligation to bond off or otherwise pay for the lien (but you would also then have a right to sue your subcontractor who didn't properly pay its sub), but it depends on the language of your contract with the GC. 

If the supplier does not file a lien, then the supplier likely only has any recourse through a breach of contract claim with your subcontractor, and that breach of contract claim would not extend upstream to you.