Unfortunately "new version" doesn't really tell us much, unless you simply meant that you rebuilt the same version with debugging symbols this time.
As I already somewhat expected from the list of loaded DLLs, since no ATI .dll is loaded, it probably means that either no ICD is installed or if it is, then it's not operating correctly on the system in question. It isn't a matter of replacing a card, just installing the proper driver and making sure it actually works, which is done in just a matter of minutes even by a layperson. And if your IT people assert that the driver was installed correctly, then ask them how they tested it, because not many people know that even in the absence of an ICD, Windows can still run DirectX games/applications because it ships a functional DirectX driver and a non-functional OpenGL driver by default, politics as usual.
If there is no ICD installed, Windows will just provide OpenGL 1.1 functionality through a Direct3D wrapper, which is probably what is happening to you. glBlendEquation is an OpenGL 1.2 function, so trying to call it when it is not provided will simply crash the application. SFML made no checks for this in the past, but recently a patch was pushed that at least emits a warning to the console if a pre-1.2 OpenGL context is created which leads to required function pointers not being loaded.
You can check the version of the context that gets created by inspecting the return value of window.getSettings(). It is more than likely to be 1.1. As long as you don't try to draw anything in this case, nothing should crash on you, but that is also pretty much all you will be able to do on those systems.
We live in the 21st century, installing the proper drivers should be a non-issue. Obviously Microsoft couldn't care less if you aren't able to run OpenGL powered applications, so your IT people would be doing them a favour by putting up with the current state of the system in question. If you explain this situation to them and they tell you to use DirectX instead, do me a favour and kick them for me... kick them really hard, multiple times if you feel it is necessary.