If you hit sell on a vanguard ETF and it sells on the market, then Vanguard isn’t the buyer is it? So in that situation with everyone dumping ETFs there would be a lag on the time taken for the ETF to sell and Vanguard to then dump the stocks back out in the market. It’s never occurred to me the situation where huge numbers of people dump index funds and how Vanguard/Blackrock account for that without becoming bag holders of the underlying stocks themselves.
In any case, I’m not sure that large enough numbers of ETF holders are sitting close enough “to the button” to hit sell in the event of a sharp downturn occurring over the space of even a week or two. And a lot of them would see it as an opportunity to DCA into the dip anyway.
If it's an ETF it's a little complicated. The usual mechanism for selling an ETF is that there's a buyer on the other end who's buying shares in the ETF itself, not the constituent stocks. Arbitrage keeps the price in line with the index constituents; if the ETF diverges from its constituent assets, some HFT can buy the ETF and sell the constituents and that will force them to converge.
However, most ETFs are also setup such that they can create or destroy shares in response to large shifts in demand. In this case, if enough people hit sell, the ETF itself will buy back shares and use the proceeds to sell the underlying assets, in a transaction that mechanically should be market-neutral and just propagate the supply/demand of the fund down to the individual stocks.
With Vanguard specifically, it's even more complicated, because VTI is not a separate ETF. It's a share class of the Vanguard Total Stock Market Index Fund. But the mechanism is largely the same - it has the same Authorized Participant system to mint new shares in case of high demand and redeem shares if everybody sells, and then passes these requests on to the underlying mutual fund, which can then piggyback on some of the tax efficiency benefits of the ETF.