Is the situation where one specific person knows that the company needs this specific person a reason worthy enough of a raise?
Worthy enough is a matter of opinion.
It's probably worthy enough from your point of view. You feel you are worth more money, so you can ask for more.
But from the employer's point of view, it's probably a bit more complex.
Nobody is indispensable. Everyone can be replaced. If not you, they will find someone else.
Companies don't have unlimited budgets for maintaining their website. At some point, you will be overpriced, and they may choose to find another way to get it done. Your challenge will be to request enough that the company is comfortable paying you, rather than going elsewhere.
I worked for a small company that had a proprietary system which was maintained by one individual. He was confident that only he could do the work. Unfortunately, he had a heart attack and couldn't work any longer. The company went to a local university and hired an intern who taught herself how to handle the system at a fraction of the cost. It wasn't immediate, there were a few glitches along the way, but in the end she was far better than the original sysadmin. And she created enough documentation that it was never a one-person-system again.
if they don't get me hired, their website won't be maintained.
No. They'll just find someone else to maintain the website if they want it maintained.
Or, as @Areks correctly points out, they could hire another company or individual who will port the website to a newer, better documented, well known technology.