This should be a system level thing, not a website level thing.
Considering that outside a handful of specialist or language sites - the use for this is somewhat minimal, and outside the Russian/Portuguese SO site, various language sites and to a small extent judaism (where questions are in english, but there's a definite need for some content in other scripts) - english is the primary language of the network.
In addition, non latin languages are complicated. There's different scripts, RTL and LTR writing, synthetic languages (Quenya and Klingon for example might belong in literature or movies) not all of which have unicode support.
Moderation is another issue. I'm not going to be able to tell the difference between poetry and toilet scribbles in a language I don't get, and I might not be able to grab the right person to tell me what's the difference.
And well, even where non english languages are supported, in the UIs I've never seen sites have localised keyboards for their own scripts. My dad runs facebook in tamil, and at one point had google in tamil - which had me translating terms back and forth, and mostly being confused.
Creating such keyboards the right way... would need developers who are intimately familiar with the language and online culture of the place. Google has 5 tamil keyboards on their input options for example -so picking what to support is going to be a pain. You'd also have trouble checking if output is correct.
This is going to be a massive amount of work, for almost no real benefit, even if you prioritized developers to do such things.
And well, the work's mostly done elsewhere.
I've found that google has a robust set of tools for android and Wintel PCs (and apparently a chrome browser plugin) - My mom used their transliteration keyboard for quite a while on windows, and uses it on android as well.
It also covers a small subset of indian languages (though presumably the 'common' scripts). I'm pretty sure most languages should have native keyboards or trivially installable support.