In most cases I can think about right now, the vocalist is writing all song lyrics exclusively. This seems to be the standard case in vk (and j-rock?).
But there must me cases where other members write lyrics as well. How common is that, which examples do you know? Is that diffrent in other scenes or genres?
Right out of my head I recall S1TK writing the lyrics for ゆらり in Ashmaze. (though I just learned about it by Azavana).
There also are some songs by DaizyStripper, but I need to look up which. I‘ll come back for that.
Nightmare is a fantastic example! The World and Alumina were written by Ruka and Sakito, the drummer and guitarist, respectively.
A huge number of their songs are written by Sakito and Ruka, but there are also various gems from the other three as well (Ni~ya the bassist has a few sleeper hits imo!)
Last year, the EP Root25 was written by each member.
Actually, in my experience, the guitarist is quite frequently the songwriter/composer
Mucc has many songs with lyrics either by Miya or Tatsuro and some random by Yukke.
Mana and Közi wrote lyrics for the final Malice Mizer album since they did not have an official vocalist at that time. Also I remember that Art Cube had their bassist N write most of their lyrics. It seems that Z/Seth/Seiji mostly did not bother writing anything much during his career up until now. He has one lyrics credit in his first band AFTER IMAGE that I know of and of course I do not know about his solo project(s). He seems to be focusing on singing as well as visual work.
I know Ayaha from DazzlingBAD wrote lyrics on few of their songs (mostly songs from their album Immortal emperor)
im pretty sure sena writes jiluka lyrics and also leda did for deluhi
That explains a lot 
I know that Miyavi did at least some if not all songwriting back then for Dué le Quartz.
Lyrics guys, lyrics!
I think Karyu wrote the lyrics to “in vain” by D’espairsRay, but I can’t find a source for that right now.
So, Yusuke wrote lyrics for lynch. - A FOOL
Here I come 
Spotify lists it for one: link
I’m seeing Forbidden, in vain, the world in a cage, but it seems to only list it for Coll:Set and Born.
There’s another entry for Karyu that lists him as writer for Subliminal, Closer to Ideal and dope.
I could cross-check with booklet scans I might still have, but that would take some time 
Edit: apparently also 月の記憶-fallen- according to my coll:set CD. For the lyrics of Tainted World it just says D’espairsRay, so maybe it was group project lol
Yoshiki is an obvious example, he wrote most of the lyrics for X Japan.
In DIV all members write lyrics from what I remember. At least that was the case before they disbanded and came back. I don’t know about now.
In UNiTE. the band member who composed the song most likely did the lyrics too.
Here I am being back for the DaizyStripper songs.
Afaik all lyrics are written by Yuugiri, except for the following ones.
Nao: Purple Candy (from BLESS) and [eating apple] (from MISSING)
Mayu: iro (from INFINITY) and 白雨、白日夢 (from FUJIYAMA)
Rei: コスモス (from シン世カイ)
Kazami: 星空と君の手 (from CROSS) and Shooting Star (from 妄想日記)
There’s also トレゾア／夕霧 listed for D.A.I.Z.Y (from INFINITY), Tresor is what DaizyStripper call their fans.
There’s Gakido.
For their 言葉 album, there’s 幽 (gt.) listed for Still in the dark.
It’s the only song I know about that, but I don’t have the composers for all of their songs at hand.
黒猫
Their Tunecore states for all songs from 彼岸 that everything has been written by ヒナタ, including lyrics.
The same applies to 少女の悪夢 濫觴 and でたらめ, the page for 夢落アリス doesn’t work but I’m positive to remember it’s the same with this song.
In cali≠gari, Ao almost always writes the lyrics for the song he writes, in fact, the first time any of their vocalists wrote lyrics to one of his songs was in 2021. He also co-writes with their bassist, and there’s a few songs their bassist wrote by himself (but usually Ao or A Shuuji writes the lyrics)