That is basically correct. Economic theory says that increased demand will lead to a shortage and that firms will react by raising their prices.
The way that this is likely to work in real life, though, is that the owners of stores will look and see that this particular product is flying off their shelves faster than they would have expected. When this happens, they will think that they might well be able to sell just as many at a higher price.
But economically speaking, that is exactly what happens -- there is a temporary shortage resulting in a higher price.
In economics when we speak of short term as a period during which the market equilibrium is in the process of being stabilized after it has been disturbed by changes in one or more variable impacting the equilibrium. Depending nature and pace of changes in the market variables, the short term effect may result in a perceptible short-term period of short or excess supply. For example, demand of candles in a city may surge suddenly due to a major breakdown in electricity supply expected to last for more than a day. In this case it is quite likely that many shops stocking the candles may run out of stock. Also sensing this kind of shortage sellers may also increase prices of candles. However, if demand for candles rise because there is an increasing trend of people lighting candles in their homes for decoration or other similar purposes, then the demand will rise slowly and there will be no perceptible short term inequilibrium. Of course it is quite possible that it may not be possible to increase the supply indefinitely even in the long-term. For example space available in a big city cannot be increased too fast and even then there is a limit on the maximum space that can be created. This is the reason we often speak of shortage of space in big cities, which results in very high real estate prices there.