Thanks for reporting this. IME-based input is super important to me -- we only speak Japanese at home. I've filed a bug [1].
I manually tested on each of Windows, macOS, Linux, and iOS (where I've done a lot of work specific to CJK input) and was able to correctly input Japanese, Chinese, and Korean text [2], so looks like this is likely an issue specific to Flutter's web runtime. I work on Flutter's desktop embedders, but if there isn't someone who's a heavy IME user on the web team, I'll gladly give em a hand to help get this fixed.