No, but it does need to be distributed in such a way that users can replace the ffmpeg provided with the product, with their own version of ffmpeg (and of course, provide a source from which the distributed ffmpeg can be built, unless it’s already configure/make/make-install upstream. Some say even if already upstream)
But also, depending on codecs, it might be GPL; e.g. x264 itself is GPLd as are some other codecs. There are BSD licensed codecs. As ffmpeg relies on many other projects to provide codecs, it’s on a codec by codec basis - though the original configure makes it easy to disable all GPL upstrrams.
The wasm port of ffmpeg that I'm using isn't modified so there's nothing for me to open source. Once I finalize my modifications of the library, I'll be open sourcing it.