You can probably work around it by booting into safe mode: that disables user-installed apps, so I'd expect you could use the Play Store without interference in that mode.
But the interesting thing here is that it's on both phone and tablet. That suggests that it's something you have installed on both, which must narrow it a little. Have you tried using some permissions explorer to identify apps that have this capability (I have Addons Detector, but many security apps can also do this)? It's also very, very unusual for an app with this permission to do this, so assuming that it isn't some adware thing I might think about whether any of the apps with it might legitimately be using this frequently/constantly (i.e. "what does app X need this permission for?"). You might try some system monitor to show which apps are active, to see whether that narrows the list. But unfortunately if there is nothing visible to give you a clue it may well come down to trial and error, in which case it may be stop one, see whether that helps, if not check it is still stopped (if it just restarts after you stop it you won't know whether it was the culprit or not), and when you have eliminated one app move on to the next.
To be honest, once I was sure of the culprit I wouldn't just stop it, I'd uninstall it. I've never met this in more than a decade of using Android, so either it is a badly buggy app or a malicious one - if it's only affecting the Play Store I would not rule out malware.
