Participating in the secondary market (Stock market) doesn't give the company any money, you just have a stake of the company that varies in price according to the demands of others and the perceived value of the company.
As far as I know there aren't any firm plans for a SpaceX IPO
Buying stock drives the price up of said stock, allowing the company to reap more money per share when it wants to sell more shares on said secondary market, and also allows it to get better deals on loans and future funding rounds. It might not directly give money to the company, but it's definitely an action anyone can take to "vote with your money" that actually benefits a company.
SpaceX remains private. Purchasing stock will be limited to accredited investors, and limited in quantity to 500(?) before the company must become publicly reporting, which carries its own compliment of pain - the new bill will raise that to 1000 I believe - still a minuscule number for a multi-billion dollar effort.
The recent launch success makes the likelihood of it going public soon a little higher, and then everyone will be able to participate - but I can only imagine all the caveats in the filing documents, and frankly after watching the debacle of a meltdown called Facebook, I wouldn't trust anyone on Wall Street to properly bring this to market unless a Dutch Auction was used.