Here are two options you could look at.
1. Have you tried updating the Google Play Store app?
Open the app, tap Settings, scroll down to Build Version, tap to enter. If there is an update option select it and allow it to update. Try to download an app and see if it works.
If it says it is already up to date, try the following: From Main Menu on tablet go to Settings > Apps > swipe right to All > scroll down to Google Play Store, tap to enter, tap clear data. Try to download an app. Hopefully this may help you.
2. Not quite sure where it is located in your Tab but go to Menu > Settings > Apps > swipe right side to ALL Apps, then scroll down and find Download Manager. The Apps are listed alphabetically, with working apps first then disabled apps.
Tap to enter Download Manager. If there is an Enable button tap it then try to go online.
If it says 'disable' tap to disable, then clear cache. reboot Tab then navigate back to download manager and enable it as above.
See if you can access the internet with your browser then try downloading an app from the Play Store.
Hopefully this is of some help