I can purchase stocks at LCN website but the BAWSAQ is still under maintenance, I've completed 45% of the game and can't buy stocks from that website. Am I missing something? Do I need a special mission to unlock it?
BAWSAQ is influenced by the online community and it's likely down due to server loads. Games as popular as GTA V almost always have connectivity issues the first week or two after release so it's probably related to server load issues. Rockstar is also aware of the issues with BAWSAQ and you can check this article for updates it's current status.
Once it is is back up, if you need any help with the stock market here's a guide I found useful.
You have to sign into the rockstarclub for the BAWSAQ to work. That's it
I'm only providing what I know, but I've had the BAWSAQ available since the start, I've even purchased shares in it already (in preparation for online becoming available). I don't even think I'd linked up by Rockstar Social account by that point but maybe I had. So long story short, I think it should work for you just maybe you keep catching it on a down time. I know I haven't been able to get SnapChat working because it was having problems, same with flight school leaderboards.