There are different types of megaphones - you are speaking here of a non-electric acoustic horn that amplifies a voice. Electric megaphones or amplifying music would be very different (but are explicitly addressed by R Rozen below)
I didn't find responsa formally addressing megaphones. However here are relevant halachot which relate to your question (taken in part from YUOnline and Zomet)
- The Gemara, Eruvin 104a, notes that there is a prohibition against engaging in an activity for the purpose of producing sound on Shabbat. The Gemara records a dispute as to whether this prohibition includes all forms of sound or whether it is limited to music. Most Rishonim are of the opinion that the conclusion of the Gemara is that only music is prohibited. Rambam (1135-1204), Hilchot Shabbat 23:4, also follows the opinion that the prohibition is limited to music. He explains that the prohibition is a rabbinic prohibition based on a concern that it may lead one to fix a musical instrument
- R Shlomo Zalman Auerbach (in Kovetz Ma’amarim, p. 41) writes that it is true that it is forbidden to make a voice heard on Shabbat by means of an instrument specifically made for this purpose, such as a record player; but the reason for this prohibition is that turning on the instrument on Shabbat is forbidden. If, however, the device was turned on before Shabbat, we have not found that the Sages forbade one to speak or sing only because his voice passes through the device and is then heard from a distance
Writing in a different but related context (the use of microphones on shabbat), R Yisrael Rozen from Zomet (the Institute for Science and Torah) writes about different factors enabling the use of some microphones in some situations which could be relevant here, e.g.,
- One may not hold the microphone in his hand; it should be set up as a
- The system should be clearly labeled as operating in accordance with Halacha
- Use of the microphone is restricted to mitzva cases, and only where there is a great need
- One may not play music instruments over a microphone, even by means of a radio set on a timer, etc.
Rav Shaul Yisraeli agreed with the above criteria, so did R Chaim David Halevi (Chief Rabbi and Chief Rabbinic Judge of Tel Aviv) and R Pinchas Baruch Toledano (a dayan in London)
So it might be that using a megaphone can be justified in certain circumstances, e.g., if held in a fixed position (and not by hand), if it cannot be repaired or adjusted, if all know that it is approved and if it is for the sake of a mitsva (e.g., to amplify a sermon in a synagogue). It is a bit puzzling why it is not more commonly used so there might be additional reasons to forbid it.
As always you need to CYLOR for a specific ruling.