I'm searching for a website search tool that inspections domain availability and integrate such tool with my website, how's this done programmatically?

You can test this domain search tool known as Domain Search Guru, it inspections domain title availability, get domain ip, location, recognition rank, whois data, snap shot.
It offers an Domain title availability API for integration together with your code.
Additionally, it includes a opera addon tool that allows you search from opera.

My personal favorite domain search tool right now is: http://instantdomainsearch.com/ Searching a lot of domain names rapidly. The outcomes are extremely snappy.

For programmatic access, you may consider DNS searches or whois searches around the domain names.