The computers can run a web browser - I used to - but they may have difficulty with the more modern later versions. If you can restrict them to IE6 (yeuch) or an elderly version of FireFox, then they will be fine - many, many computers of that spec (or much lower) browsed the net in the early 2000's.
The RAM is the killer for .NET - In theory .NET 4 will run on a 1GHz processor, with 512Mb of RAM:
MSDN, .NET system requirements[
^] - but I would take these with the same large pinch of salt that I use for PC Game Minimum Spec requirements...
You could try it. Wirte a quick 'n' dirty .NET 2 app, and install it on a target PC. See if is works at an acceptable speed. If it does, then great! If not, then VB6 or C++ native code are going to be your best bet.
One piece of advice, which you are going to hate: Do your development on one of the restricted machines. If the app speed frustrates you on it, then it will be a serious PITA for the poor user who has to sit there using it for eight hours a day. If you develop this on a fast machine, you will not realise when you have just implemented something that slows it to a crawl...
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