I've NC for this (was due a change anyway!)
I make a living as a writer, and I'm happy with it, but I do a lot of work to commission rather than just sending off my own manuscripts IYSWIM. I can do that because I've built up contacts and a reputation over a long time. It's hard work, I'm self-employed, have a lot of deadlines and there is no option to have writers' block or wander around being inspired :o - to make a living I have to be efficient and organised. It suits me because it's flexible with having DC and I enjoy it a lot, but with my education and skills I could have done something a lot better paid. It's high-status and impresses people, and of course rewarding, but not so much financially!
(On the plus side there's very little to invest, you just need a computer)
With children's picture books, very few people do that as their career and don't do anything else. It's possible if you are very popular and prolific like Julia Donaldson, or have a massive hit like Guess How Much I love You etc. But generally picture book writers will be doing it part time and working in creative writing teaching, doing a lot of school visits, or still have a day job of some other kind.
Be aware that most writers earn way less than average, in surveys writers' earnings from writing averaged below 11K a year - that's why they do other jobs. You hear a lot about rich and famous writers but that skews the reality.
With picture books, what you need to do is joins some kind of writing group or course, an Arvon course would be ideal - then you can get professional advice on what you have so far, how good it is and how to polish it up. Then get the Writers' and Artists' Yearbook which has lots of advice on how to submit your idea or manuscript, and lists of publishers and agents so you can find the most suitable ones to send it to.
There are also some publishers who have annual competitions for new picture book authors - the Yearbook should help with this. Also google for writing organisations and websites where they list things like competitions and calls for submissions, and have helpful forums etc.
You could strike it big but you have to plan for it always being a sideline, IYSWIM.
There's also self-publishing / e-publishing and that can be a success, so look into that.