Can I sue my insurance company if they refuse to compensate me for my stolen vehicle?

You can try suing them for "breach of contract"--for violating their contractual (an insurance policy is a contract) obligation to pay for your loss--but you'll have to provide this same information in court. The insurer is allowed to investigate claims, to make sure they are legitimate: in the context of a car theft, that often means investigating whether it may have been insurance fraud: seeing if the owner either paid someone to steal his car so he could put in a claim, or "sold" his car under the table and is now claiming for it. That involves looking at phone records (who have you contacted) and bank records (unexplained transfers of money or checks), among other things. If you sue them, they will defend on the basis that they were simply doing their legimitate investigation and will furthermore use the lawsuit itself to demand this material.
In addition, you have an obligation under your policy and under the law to cooperate with your insurer in investigations; a refusal to turn over material they requested will entitle them to deny the claim. Your only way of winning, were you to sue, would be to simultanouesly show that the materials they sought was unreasonable or unnecessary to the investigation (though it appears like it was reasonable, as discussed above) and that you did not in fact commit insurance fraud (since even if they asked for too much, if you did commit fraud, you are not entitled to compensation).