As a non-academic, how do I find papers that reference a certain article?

Google Scholar can get you quite far.

Go to scholar.google.com and paste in the full title of the paper you're looking for. Here's one of mine:

Click the "cited by" link (highlighted) and you'll get a list of works citing the paper you're interested in. Some of these will be papers, some won't, and Google's indexing isn't quite the same as some of the other sources. But it's close. For each of the citing papers, there's an "All n versions" link. Browsing those versions will often get you to a legitimate copy from the author's institutional repository (see the right-hand column), ArXiv, or various other sources; the Unpaywall browser extension can also help track these down.

Zentrallblatt https://zbmath.org/ allows (limited) free access and if you start with the paper that you are working from, you might be able to get some information. Of course, full access to Zentrallblatt or MathSciNet is what you need but it requires a subscription. If you can go to a University library, you can explain your situation to a librarian and they might help you. If you know someone who is a student or works at a university, they can do the search for you.

Edit; You should also try Google Scholar.

An alternative less onerous to the author is to send him an email with the statement of your result and ask him if he's seen it before.

Many academic libraries allow outsiders to go there in person and use their search facilities.