Yes, it is possible for a stock to have no buyers. Typically, this happens in thinly-traded stocks on the pink sheets or over-the-counter bulletin board (OTCBB), not stocks on a major exchange like the New York Stock Exchange (NYSE).
When there are no buyers, you can’t sell your shares—you’ll be stuck with them until there is some buying interest from other investors. A buyer could pop in a few seconds, or it could take minutes, days, or even weeks in the case of very thinly traded stocks. Usually, someone is willing to buy somewhere: it just may not be at the price the seller wants. This happens regardless of the broker.
The broker only places your order in the marketplace so it can transact with other orders. The broker itself does not typically try to solicit a trade in a stock, which means your decisions to buy and sell are up to you, and the broker just facilitates those decisions.
If an institution acts as the principal to a certain amount of stock, a rapidly declining stock price will affect them. This is because, unlike an agent, the dealer is an owner of the stock. Examples of this include market makers.
*The above answer is borrowed from Investopedia