From Wikipedia I can see that an Esperanto flag exist.
But how about the Latin language? A possible use of the flag could be at language learning sites like Duolingo, where other languages are represented by country flags.
And who (which organization) could decide on this issue if an official flag should be made?
Very few languages have flags per se. For example, the Finnish flag stands for Finland, a constitutionally bilingual country, not the Finnish language, which is also spoken outside Finland to some extent. There is no flag for the Finnish language. Nor for English as far as I know.
Some languages can be easily identified with some flags; consider Italian, French, Greek, and German, for example. But the languages are not confined to their main countries, nor are they the sole language in them. Although often innocent and sometimes useful, identifying a flag with a language can be seen as a political statement. A Canadian might be upset that their native language is identified with a foreign flag (French), or a Swedish-speaking Finn might be upset that the flag of their fatherland is identified with another language (Finnish).
If Esperanto has a flag dedicated to the language, it is a rare exception. Flags are typically for countries and other administrative entities, not languages.
Some languages can be easily identified from a flag in contexts like language learning software or choosing the language of a website. I would argue that Latin is not.
The proposed Latin flag linked to in the comments combines Rome and Vatican. However, both Rome and Vatican operate mostly in Italian today. The role of Latin in Vatican is mostly ceremonial. Although Vatican is the country most strongly linked with the Latin language, Latin is (in my opinion) not the language most strongly linked with Vatican. Some may argue that the Vatican flag stands more for Latin than Italian; perhaps it does, but the ambiguity is inevitable. If the purpose of a flag is to make the user interface better, ambiguity is certainly something to avoid.
As Rafael commented, perhaps the best symbol is the Vexilloid of the Roman Empire (image from Wikipedia):
This corresponds quite uniquely to Latin. However, I see two issues:
Anybody can decide to issue a flag for anything. Perhaps I could draw and publish a flag for mathematics. Such declarations will have little effect unless done by a suitable body. Latin spans many eras and areas in Europe, and no single institution can claim it to itself. It might be taken seriously if Vatican or ALF issued a flag, but it is unlikely to catch on unless several organizations and individuals agree to it and start using it. A flag without public awareness and support is quite useless, and no person or institution has sufficient power over Latin to do anything about the language unilaterally. There is no central authority.
Any institution would need a strong reason to introduce such a flag. Adding a symbol retroactively to something of such a great historical importance would easily be frowned upon. I believe I am not the only one who will not be easily convinced to accept a flag for Latin.
@Joonas Ilmavirta is right, there is no official flag per-se for Latin or most languages for that matter, but this answer addresses if you were looking for an icon to use to represent Latin, e.g., on Duolingo.
The flag used to identify a language typically depend on who the users associate the flag with. For example, a website used primarily by Americans would use the U.S. flag to indicate English, while a site used by Europeans would probably use the British Flag.
The big question to figure out what flag to use in practice is who the users are, whether they're primarily interested in classical or ecclesiastical Latin.
The Latin Catholic Church has been using Latin now for a few centuries longer than ancient Rome did, so for most cases the flag of the Vatican City State is the best choice to indicate Latin. Most people associate the Catholic Church with Latin first, even if the primary operational language of the modern Vatican City State is Italian. The Vatican flag should certainly be used if the ecclesiastical pronunciation is used, or the thing you're using the flag for is somehow linked to the Church.
For things that focus more on the Latin of ancient Rome, especially if it uses the classical pronunciation, the Vexilloid @Joonas Ilmavirta suggested would be best. Users looking for the classical Latin will in all probability be able to distinguish between the SPQR and a Nazi banner.
Duolingo has just released Latin as a language to learn. Here is the flag they ended up using:
Here is how it looks in the app:
