Recently been asked to try and sort one of these out. It has a two fold problem, it won't access Google's Servers when trying to run Play Store and secondly it won't install anything using an .apk file.
I've reset it to Factory defaults deleted the Android Secure folder it's not a WiFi (my own Galaxy tab 2 has no access issue on the same connection) issue it has no trouble accessing the Internet including Google and Gmail the log on email is valid I've trawled the net trying to find an answer but nothing out there seems to work .
I don't like giving up - but it's reaching the point where I have no option.

You normally have to change a security setting to allow sideloading of apks.

No idea about google playstore but here are some suggestions. I expect you have tried them but just in case.

https://www.androidpit.com/google-play-not-working

I'll try again later, but having tried so many things including a factory reset I'm beginning to think there's a serious fault or corrupted system file.