Is there a way for a mortal to buy or invest into SpaceX or are we just waiting for an IPO eventually?
Go work for them on a position that has equity as part of the compensation. Otherwise, I doubt it. Elon has been pretty vocal about regretting making Tesla public. Additionally SpaceX as a company is oriented around putting humans on mars. Until that happens, I expect a supermajority of ownership would be against the profit seeking motives that an IPO creates. Short of investing in them, you could look for companies…
Talk of putting people on Mars is how they recruit young idealistic engineers. I think to understand what SpaceX is actually about you need to understand who Michael Griffin is and what he's all about. He went with Musk when he tried to buy an ICBM from Russia; he conceived of NASA's Commercial Resupply Services program and gave the contract to SpaceX, before SpaceX had ever put anything into orbit. Ditto the Commercial Orbital Transportation Services contract. It's not an exaggeration to say that SpaceX would not exist if not for Michael Griffin.
Read about the rest of his career and I think you'll figure it out: https://en.wikipedia.org/wiki/Michael_D._Griffin#Career