If while working on a construction jobsite with the company van and getting tools from it, the door blew open and struck a neighboring vehicle, am I responsible?

You are both right and wrong.

Your employer's insurance probably does cover damage done to another employee's vehicle by a company vehicle during work--it is impossible to say for certain, however, without reviewing the policy, since an insurance policy is a contract, and an insurer is only obligated to pay when the conditions or events set out in the policy occur.

However, if you were careless, or negligent, in causing the damage--for example, you swung the door too wide, or failed to hold onto it to prevent it from being blown open--you would be liable for the damage you negligently caused. The co-worker could sue you directly for the money; and even if the employer's insurance covered the loss, the insurer would then have the right to sue you, as the at-fault party, to recover the money they were forced to pay due to your carelessness.

So it may be the case that the insurance should or will pay, but you will also be held liable for the loss.

Also, do not forget that unless you have an employment contract, you are an "employee at will" and may be fired at any time, for any reason. For example, your employer could legally fire you for damaging the co-worker's car then not taking responsibility for the damage. You need to consider this in deciding whether to pay the loss or not.