I'd recommend having a look at the OSGeo site. They have loads of information about open source GIS from all parts of the spectrum. Look at the right hand side of the home page and you'll see 'Web Mapping' which lists a bunch of Open Source applications you can use for web mapping.
At my organization we personally use (which many others also use) GeoServer for a the server side WMS/WFS etc. and OpenLayers for the clientside functionality.
This is more if you or someone else were willing to develop an application around it, but there's loads of good tutorials out there and it is far less daunting than it first may seem. There would also be a fair few companies who could build it all for you at a far reduced cost than an ESRI License, but you'd have to do your own research there.
I hope this helps if only to make you aware of the mass of open source solutions out there. Just for reference, this is the application we made with GeoServer, OpenLayers and VB.NET (You can use whatever server side programming language you want) - Dorset Explorer