Getting redirected to login page on every other CP request when using Vagrant
I kind of doubt this is it, but Craft does have a couple of config settings that might be coming into play here.

You can tell if you're being affected by these by looking in your craft/storage/runtime/logs files and see if there are any messages being logged with a [warning] related to not being able to restore session.

If you are being affected by those, try setting both of them to false just to see if that helps.