How do I find an older version of an app?

Because I have an older iPad that I just inherited, I can't run important apps now that the new iOS update has come out. Does anyone know if there is a way to get older versions of the apps from apple? I have already checked with the developers, who say that apple should offer an older and compatible version of their app. But, when I got to download them I just get the message that I need to update to ios 6.1.1.