Yes, of course. Supply and demand are at the heart of Austrian economics. The solution Dr. Paul proposes here basically amounts to "stop the government interfering with supply and demand." Essentially, the shortage here is resulting from the government preventing price from rising to reflect supply.
Right now, what is in place is called a price ceiling. A price ceiling is a government initiation of force that sets a maximum price and which a product can be legally sold. What basically happens is that the ceiling prevents the free market from reaching its natural equilibrium price and quantity. The equilibrium is basically where supply and demand are equal, and therefore everything that is produced is (in theory) consumed. When a price ceiling keeps the market below the equilibrium price, this causes the quantity demanded to exceed the quantity supplied. This results in a shortage, which is why we were seeing lines at gas stations.
The price ceiling can basically be illustrated by this diagram: http://img.sparknotes.com/figures/0/039bab1e6f1ef2a65b5f4c8d...
The blue and red lines symbolize supply and demand, respectively. The point at which the lines cross is the natural market equilibrium, where everything produced is consumed, and there is no excess supply or demand. The black horizontal line that crosses the supply and demand lines symbolizes the price ceiling. You can see that on the horizontal line that is the ceiling, the demand is further to the right than the supply, which illustrates demand exceeding supply. The distance between the supply and demand curves on the line of the price ceiling is the amount of shortage.
How do we solve this situation? Well, as Dr. Paul says, price ceilings (as well as their counterpart, price floors) should be ended. In this scenario, the price of gas will naturally rise to the equilibrium. Since the price is higher, the will be less demand. This will get rid of the lines. Since gas prices might go up to, say, $15/gallon (just making up a figure, I don't know what the natural price would be in this particular situation), people who do not necessarily need gas will not buy it, but those to whom it is important will pay the price. Yes, not everyone who WANTS gas will get it, but the same is true with the government setting price, except that in the authoritarian version, the gas goes to those who get to the station first, while in the capitalist version, the gas goes to those who are willing to pay for it. In the opinion of libertarians, this is more fair than the controlled system. Also, it does not require violation of property rights of gas sellers. This is true without even MENTIONING the fact that few companies will truck gas in to get $5/gallon, but a lot more would be inclined at $15/gallon. This would cause a natural growth in supply, which would then naturally decrease prices as the emergency situation came to an end.
I'm sorry if you already knew the stuff in this post and it came off condescending, but you asked how supply and demand apply to Austrian economics, so I felt it necessary to give a primer as to how supply and demand apply to the situation by describing the nature of free markets.
If you are indeed new to economics, I recommend having a basic grasp of supply and demand (essentially what I described in this post) and then reading the book Economics In One Lesson by Henry Hazlitt. This is available from the Mises Institue (https://mises.org/store/Product2.aspx?ProductId=33) and it can also be obtained on iPad (and probably Kindle). Hazlitt explains economic fallacies commonly espoused by Keynsians, socialists, and other forms of statists and disproves them through economic logic.