In the United States, whenever a customer places an order, will the broker always execute that trade on the market?
Not necessarily. For stock options, yes. For stock, they may be sent to an exchange or stock market for execution or they may be traded against another customer's order, against the orders or quotes of an affiliated market maker, or the broker's own account. Whatever happens, the broker is obliged to provide "Best Execution" and one of those things is ensuring that the customer does not pay more than the currently displayed National Best Bid or Offer (NBBO).
Do brokers just maintain a "reservoir" of popular stock, and hand them out to customers at market price instead of going to the exchange?
Unlikely as they are bound to provide the customers the stock at the NBBO or better, and as the stock price fluctuates they would be subject to risk. However that is not to say they might try to match customer orders against each other, and many brokers participate in "Dark Pools" (effectively non-displayed markets) that can be used to try match up orders instead of sending them to the exchange or stock market.
In other words, do people ever just buy/sell assets from their broker, instead of through their broker?
It is not that common, but yes, it can happen.
Whatever they do, the broker is required to record all of the things that happen to your order (Reportable Order Events) to the Financial Industry Regulator (FINRA) through their Order Audit Trail System (OATS) on a daily basis, which can help to keep your broker honest.