To be honest, I'm still unclear as to who everyone is in this story, but in a nutshell:
If you an an employee of a company that utilizes the services of a supplier, and part of the contract between your employer and the supplier is the supplier will obtain and maintain X amount of insurance coverage (very common requirement), with very few and rare exceptions*, you as the employee have no legal right to "force" the supplier to actually get the required coverage. That claim belongs to your employer, who would bring a breach of contract suit for failing to procure requried insurance.
Of course, that doesn't preclude you from suing the supplier anyway - it's not your problem how they pay any potential judgments.
* It would take too long to get into the exceptions, but if you're interested, do a search for "third party beneficiary texas".