Let's say I see a popular closed source application (distributed for free in binary form) links dynamically with a library that has a GPL license (according to the library's website). Can I write to the company and demand the source code to their app?
Yes, you can ask for that.
However likely only the authors / the copyright holders of the GPL library will have legal leverage as their copyright has been violated.how exactly will strongly depend on your jurisdiction. Before you go wild, you might want to check that the library copyright holders did not give a special license to the authors of that programme.
Thus I recommend to do both: (a) inform the copyright holders of the library. (b) Ask the authors of the violating programme to give you the sources.
And lobby the library authors to take action, in the same way and to take legal action, if you don't get the sources of the programme - provided that no special license was given to the programme authors. You might also inform the Software Freedom Law Centre, gpl-violations.org and/or FSF. Maybe they can support the library authors in some way in their claim. Legal action is expensive and not necessarily a thing every person wants to burden themselves with for financial or psychological reasons.
However even legal action might not mean that you get the source. Result might just as well be that the programme is withdrawn completely or simply changed to not rely on that library.
It depends. Specifically, it depends on jurisdiction. Yet again, I go back to the "Licences and Contracts" talk at FOSDEM 2018 that I wrote up for LWN. The relevant bit was what are called third-party rights, which specify the extent to which anyone other than the licensor has standing to sue a licensee for failure to observe the terms of the licence.
The opinions of the lawyers giving the talk were that: in England and Wales, no third-party rights exist. In the US, third-party rights likely depend on the precise wording of the licence (which I heard as don't count on it). But in civil law jurisdictions, such as most of Europe, the opinion of the speaker was that third-party rights do exist.
So if you're in such a jurisdiction, it's worth a try. If the company in question is also in such a jurisdiction, it's even more worth trying. Though in the end, planetmaker is right when (s)he notes that litigation is painful and really expensive, so you'll want to do more than just ask for opinions on a community board before you start throwing solicitors' letters around.