Just like how banks are backed (up to 250K per account held) by FDIC insurance, investment companies like Vanguard and Fidelity are covered by Securities Investor Protection Corporation (SIPC) insurance in the event that the company goes under.
http://www.sipc.org/Here's a decent explanation on Fido's site as they cover above the standard SIPC limits:
https://www.fidelity.com/why-fidelity/safeguarding-your-accountsYou have to rely on these type of companies be composed of mostly honest employees and that they have adequate oversight. You do your homework and research, and go with the company that you can find good information for and ultimately trust in their track-record and amount of satisfied customers.
If you're so freaked out that there are no records or stock certificates, you can request monthly statements by mail, or download them off their websites and save in your computer. Would be something anyway that might help you feel better about having concrete proof of your funds and how much/where they are.
I personally don't believe a company that has been in business as long as places like Vanguard and Fido will suddenly turn shady and steal all my money. Even if someone at the company DID do that, they'd restore every penny as soon as the fraud was discovered, and the SIPC would indemnify you if the entire company suddenly turned evil and shut down overnight, so you'd still eventually be made whole. But the idea that something like that happening is honestly in the realm of "ice skating party in hell" time - serious level tin foil hat wearing paranoia in my opinion. ;)
There is the nuclear/zombie apocalypse scenario, in which case likely unconcerned as much with the state of our investment accounts as much as wondering where we'll hide from the undead/nuclear fallout.
Otherwise, you're welcome to pull out all your money as gold bullion and bury it in the backyard... ;)