There are all sorts of "formmail" scripts available, which will email requests to you. Many have been discussed on this very board (use the search function). You would just drop a few files into your site and fill out a configuration data file. That would be the simplest, but requires that someone look over all the accumulated requests when a cat comes in, and decide who's the closest match. The next step up would be to customize the form that visitors see, with decoration and flourishes such as randomly selected pictures of cute kitties just waiting to be adopted. That would require some programming skill. The next step up would be to put the incoming form data into a database (rather than emailing it), and use the computer (on the server or on a PC) to do a rough match (narrow down applicants to a few possible candidates, to ease your burden). That, of course, would require quite a bit of custom programming, but might be worth it if you will be handling a high enough volume of animals. You might be able to recoup your development costs by selling the customized software to other shelters (at a fair price, of course). The example link you gave has lots of good data input, but is so plain as to be utterly unappealing. There's no incentive to go through the bother of filling it out. By the way, it's written with MS Visual Studio and submits the form data to an ASP.Net script, so I'm guessing that it goes into a database rather than being emailed to someone.
Have you actually searched to see what's available for shelter software that someone's already written, and providing for little or no cost? I can't believe that there isn't something suitable out there already.