3 Months seems way too long. No, you do not have to wait for the insurance company's subrogation claim to resolve. Your insurance company owes you a duty to act reasonably. If they do not act reasonably they are in breach of the covenant of good faith and fair dealing. Withholding payment of your property damage claim for this long does not seem reasonable. If the amount of property damage is not in dispute, an undisputed payment should have been made a long time ago. An attorney could put pressure on the insurance company to do the right thing immediately. However, I suspect that the insurance company will eventually pay your claim. If you can't wait, hire an attorney. If the delay in payment is causing you other problems, i.e. you are unable to use your vehicle until it is repaired, then you may have a cause of action for bad faith (i.e. breach of the covenant of good faith and fair dealing).
Yes it is legal. They have an obligation to conduct an investigation. Write a demand that they pay you within 30 days or you will be forced to retain counsel. Then follow through with your threat. Best of luck.
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As I understand your question, you have made a claim with the other guy's insurance, not your own. I assume this is because you don't have collision or comprehensive coverage on your own policy. (If that's wrong and you do have collision or comprehensive coverage, you should have tried going through your own insurance.) Three months is an unusually long time to wait. I would strongly suggest you consult with an attorney. (Most will provide free initial consultation.)
What may be happening is there may be a limited pool of money from which the other side's insurer has to dole out all property damage payments, and they are trying to figure out how to divide up that pool of money. You need someone to advocate for your interests just like the semi truck driver's insurance company and/or attorney is/are looking out for that party's interests. If your payment comes up short, you may have to resort to your own insurer for an underinsured motorist ("UIM") claim -- again, something that would be easier to do with an attorney's help. Best of luck to you.
This response is based on limited information. It is not meant as and does not constitute legal advice and does not create an attorney-client relationship.