My webmaster signed an agreement saying that all work done on the website is owned by our company.
He now thinks that he is being payed unfairly and wants the website to be taken down.
He says that he will allow us to use the code after we pay him a certain amount of money he thinks he is owed.
Is this a common practice?
If all work done on the website is owned by your company then unless there is some specific part of the contract he is quoting it sounds like he's out of luck. That said, if you are able to make a backup of any files pertaining to the website, do so ASAP.
You can try threatening him with legal action, in which case:
I also imagine that this is a common practice, known as extortion.
EDIT - This assumes no other important details. I don't know your contract and I don't know why he feels he may have been wronged. Also I am not a lawyer.
Yes, as user745434 suggests, this sounds like extortion, which is frowned upon in many jurisdictions.
Try to get to the root of why the webmaster feels bad, as this will likely be the key to solving this issue. Ask them why they feel they have been treated badly and why they didn't mention it before when the original negotiation took place.
Yopur development agreement with him governs - and it will have a provision about who owns the copyright in and to the site (and the code behind it). The problem is that you will need to access it, and if he controls access, you may have a great case legally but a tough case as a practical matter. If he were to deny you access and then put up your site, contact the ISP and send a DMCA takedown notice.