They do, that's what the collateral is for. From the linked article:
>The way that stock markets mostly deal with this risk is a system of clearinghouses. The stock trades are processed through a clearinghouse. The members of the clearinghouse are big brokerage firms—“clearing brokers”—who send trades to the clearinghouses and guarantee them. The clearing brokers post collateral with the clearinghouses: They put up some money to guarantee that they’ll show up to pay off all their settlement obligations. The clearing brokers have customers—institutional investors, smaller brokers—who post collateral with the clearing brokers to guarantee their obligations. The smaller brokers, in turn, have customers of their own—retail traders, etc.—and also have to make sure that, if a customer buys stock on a Monday, she’ll have the cash to pay for it on Wednesday.
The catch here seems to be that the broker can't use customer funds for collateral (see my other comments in this thread), so the broker has to come up with the money themselves by drawing on lines of credit. If those lines of credit run dry, then they can't take any more orders.