Hi
@striedo, welcome to the forums.
This question gets asked a lot, but you can put your mind at ease.
When you buy a Vanguard index equity fund, your money gets pooled into the Fund. The Fund goes and purchases in the correct market weight proportions, every stock in the index that it tracks.
If a stock that the fund holds goes bust, the fund loses some value, and the value of your investment goes down accordingly.
Every quarter (or whenever the index changes), the fund will "rebalance" by selling the stocks that have fallen down out of the index, and purchase the stocks that have been added to the index.
In the very highly unlikely event vanguard goes bust, the management of the fund - being the management of the computers and algorithms and the people that manage that, will likely be taken over by another provider. Could be State Street, Blackrock, Fidelity, the Vampire Squid.... hard to say who exactly, but the fund will go on.
In the event your broker goes bust, you don't lose your investment, unless they have invested your money in their name instead of yours. This could be if they've done something naughty like pool client monies and mix company money with customer money. It's a big no no, and something to watch out for.