A business can't claim it back as such. All they do is work out how much they collected from their customers, then work out how much they spent on business supplies and take the latter from the former which is what they pay the VAT man. Since the school won't be collecting VAT from anyone it's highly unlikely they can claim it back.
There might be an educational clause in the tax system regarding VAT that I'm not aware of, but I doubt it for things like paint even if it does exist especially when it's not being done by professionals (would be a pottentially nice tax fiddle for someone if it could).