I'd personally say your biggest stumbling block would be the version of Java on the school's PCs. Back when I used to support schools' IT, there was a requirement (due to other software) for Java to not be upgraded beyond 5u17 (though that was a few years ago now and is probably higher now). While it is possible to have two versions of Java installed on a machine it's fiddly and, personally, if I was asked to install the latest just so the kids could play Minecraft in their free time, then I wouldn't - but your school's IT admin might be nicer than I am.
If you could get past the Java stumbling block (and also updating the video drivers) then I see no reason why Minecraft wouldn't run on the machine - like you say it doesn't have very high requirements, especially in the lowest graphic settings.