If there were ever a situation where investors en masse dumped shares, which is possible but infrequent, then yes, market makers must take the other side of the trade:
Market makers must maintain continuous two-sided quotes (bid and ask)
within a predefined spread. A market is created when the designated
market maker quotes bids and offers over a period of time. They ensure
there is a buyer for every sell order and a seller for every buy order
at any time.
This is applicable on the NYSE as described above, and for the NASDAQ too:
The Nasdaq requires market markers to provide a “two-sided quote” in
the securities they cover. This means they must post a price they will
buy at and a price they will sell at... The market maker on the Nasdaq
is responsible for insuring a market is available for listed
securities... The market maker assures that there will be a market for
the security; however, they don’t guarantee it will be at the price
One way to think of it is that being a market maker has both risks and rewards. It is generally an easy way to make money off of the bid ask spread. But the downside is the market makers' duty to always take the other side of a trade in the stocks that they cover. This is how the NYSE Glossary defines "market maker":
A market maker must at all times display bid and ask prices, for which
minimum quantities and maximum spreads are defined instrument by
instrument. A market maker must also meet minimum volume requirements
in the contract(s) in which it makes a market. In return, market
makers pay lower transaction fees.
If a company's stock is dumped by investors, it is usually a symptom of something the company did wrong, or failed to do at all. It is uncommon for an NYSE listed firm to just have its stock dumped. Reasons for dumping might include suspected fraud, or sudden news indicating a poor outlook given the company's market.
The other problem when this happens for the company is that when the stock is dumped so dramatically, there will be suspicion and negative sentiment i.e. "when there's smoke, there's fire". It will be seen as a sign of poor future prospects. Corporate debt might be downgraded, thus negatively impacting the company's ability to raise funds through future debt offerings.