I developed vibrato myself, and was able to produce a passable sound after a few months, but it took years until I was happy with it and even then I am not sure that I am doing it "right".
I started purely because I was impatient to "sound like proper violinist". Back then I didn't even consider whether my notes were in tune or not. I had a peri teacher for a few months who never commented on such things (class of 3-4 students, 20 mins for a lesson - there was no time for finer points like that....). Then when I left school it never occurred to me

I know that is awful and I would have been playing dreadfully out of tune for years.
Anyway, regarding the vibrato, I started by doing a "finger vibrato" - just by moving my finger up and down on the string. Gradually I began to develop the wrist vibrato, until the old lady who lived next door to me (who used to play the violin) told me to "learn to do arm vibrato, because wrist vibrato sounds awful". So I concentrated on that and eventually developed an OK-ish arm vibrato. The trouble was, I couldn't control the speed of the vibrato.
A few years back I had a couple of violin lessons with a girl who was studying at the RAM. She told me that I was badly out of tune all the time and my vibrato was too stiff and that I should learn to do wrist vibrato. Then I went into hospital and she graduated and she never taught me how to do it, but I've been developing it myself since then and now I am really quite proud of it because it's something I've done all by myself. I can control the speed and intensity of it and can colour notes with it exactly as I wish to, which feels very liberating. I also worked on really listening to whether I'm in tune or not......
However! My viola teacher told me that my vibrato needs to be wider for the viola. She said "We must get you learning arm vibrato" which was nice because I can already do that! It's gone rusty though, but at least I can do it to some extent, and don't have to start from scratch. I usually play with a combination of wrist and arm anyway.
Sorry, this was a bit of an essay!