How can there be any grounds for someone to claim that a requirement to report a theft of anything at all, let alone a gun, can possibly infringe their rights or harm them in any way?

Take a look at the news article TIA linked to at 4:06 PM. It sounds like the NRA simply doesn't want local governments, Philadelphia in this case, making any firearms related laws. Pennsylvania law gives the right to regulate firearms to the state, not to counties or municipalities. Even though the law seems innocuous, even by gun-nut standards, the NRA doesn't want cities poking their noses into gun issues at all. The NRA doesn't own nearly as many politicians on the Philadelphia city council as it does at the state legislature in Harrisburg.