The act of reporting the vulnerability likely isn't the problem with the police. The fact that he found the vulnerability is probably what caused the problem the police. Using the website as it was intended to be used almost certainly wouldn't reveal an SQL injection vulnerability. For him to have found it meant he was doing something that may have been illegal.
Just because a website exists does not mean that you have the right to poke at it to find its vulnerabilities. Unless you've been authorized by the owners to conduct vulnerability testing, your actions will be viewed as malicious and may be illegal depending on where you live, and may result in serious consequences even if your intents were not malicious. Unfortunately, the days of something like this being viewed as "harmless" are pretty much gone.
In some ways, this reminds me of the "Kasper Holmberg incident" in Canada in 2008, in which a "well-intentioned" student at Carleton University identified a vulnerability in their student card system and exploited the vulnerability to access email accounts and financial information of a number of students so he could write a paper he sent to the university. He was charged with a number of violations of the criminal code of Canada, sanctioned by the university, and ultimately ended up dropping out of the university. The criminal charges were withdrawn several months later, but that doesn't change the fact what he did was illegal, even if it was well-intentioned.