Step 1. Pick an inference engine. There are many choices. Here's a list: http://en.wikipedia.org/wiki/Expert_system#Shells_or_Inference_Engine
Step 2. Use the language that interfaces with the inference engine.
You'll be much happier leveraging an inference engine for expert systems work.
I would like to put this up on a blog or website and let people use it from there
Trivial.
is there a particular inference engine that works with the .NET family, or PHP, or something to that effect?
Doesn't matter.
Here's the confusion. Your "web site" and your "inference application" have NOTHING to do with each other. Nothing.
Your web site can be done in any tool set you can find. It doesn't matter.
Your inference application can be done in any tool set you can find. It doesn't matter.
Your web site will invoke the inference application through any API that makes sense. The lowest common denominator in API's (the reason that none of these choices matter) is to do this.
Write your inference application as a stand-alone command line tool.
Write your web application to run the stand-alone tool, collect the output and turn the output into an HTML page.
Note that this multi-porocess implementation may be faster and make better use of multi-core processors. It forces the OS to manage the web server (Apache HTTPD, for example), the web application and the expert system as potentially three, separate, parallel processes.