Some people believe that inflation is caused by an increase in the money supply when the banks engage in fractional reserve lending. Is this correct?
You are referring to the Austrian school of thought. The Austrians define inflation in terms of money supply. In other words, inflation is defined as an increase in the aggregate money supply, even if prices stay the same of fall.
This is not the only definition of inflation. The mainstream defines inflation as a general increase in the prices of consumer goods.
Based on the first definition, then your supposition is correct by definition. Based on the second definition, you can make a case that money supply affects prices. But keep in mind, it's just one factor affecting prices. Furthermore, economics is resistant to experimentation, so it is difficult to establish causality.
Austrian economists tend to approach the problem of "proof" using a 2-pronged tactic: establish plausibility by explaining the mechanism, then look for historical evidence to back up that explanation.
As I understand it, when there is more available money in the market, the price of goods will increase. But will a normal merchant acknowledge the increase of money supply and raise prices immediately? I posit that, in the short run, merchants won't increase prices in response to increased money supply.
So, why does increased money supply lead to price inflation?
The simple answer, in the Austrian school of thought, is that you have more money chasing the same amount of goods. In other words, printing money doesn't actually increase the number of widgets made.
I believe the Austrian school is consistent with your supposition that prices don't increase in the short run. In other words, producers don't increase prices immediately after observing an increase in the money supply.
Specifically, after the banks print more notes, where will the money be distributed first?
The Austrian story goes as follows:
Imagine that the first borrower is a home constructor, and he is borrowing freshly "printed" money to build new homes. This constructor will need to buy materials and hire labor to build homes, and in doing so he will bid against other home constructors. The increased demand for lumber, nails, tools, carpentry, etc. will ever so slightly increase the market prices for these goods and services.
So the money goes first to the borrower, but then flows also to the people selling to the borrower, and the people selling to the sellers, etc. It has a ripple effect.
Who will be the first one to have a need to rise their price?
These producers won't need to increase their price, but they will choose to do so if the believe that demand outstrips supply. In other words if you have more orders than you can fill, then you may post higher prices because you think consumers will tolerate the higher price.
You might object that competition deters any one producer from unilaterally raising prices, but in fact if all producers are failing to keep up with demand, then you can unilaterally raise prices because other producers don't have any excess inventory to undercut you with.