Can I file suit against a gun dealer who didn't secure guns in his shop so they were stolen and one ended up being used to shoot me during a burglary?

Unfortunately, probably not. That's because the law does not generally make person 1 responsible for the criminal actions of person 2,  even if person 1 used something belonging to person 1 to commit the crime. You would have to show that the likelihood of a theft was particularly high or likely--and more; that the likelihood was that the guns would be stolen by someone who would use them in a crime or sell them to a criminal to use in a crime, and not by collector--to have a chance of establishing liability.