Hello and welcome to Apple Support Communities, rasmus02!
I understand you're unable to turn on data for the App Store. I'd be happy to help you troubleshoot this issue.
Make sure you don't have restrictions enabled: Use parental controls on your child's iPhone, iPad, and iPod touch
"Allow changes to other settings and features
You can allow changes to other settings and features, the same way you can allow changes to privacy settings.
- Go to Settings and tap Screen Time.
- Tap Content & Privacy Restrictions. If asked, enter your passcode.
- Under Allow Changes, select the features or settings you want to allow changes to and choose Allow or Don't Allow.
Here are some features and settings you can allow changes for:
- Passcode Changes: Prevent changes to your passcode.
- Account Changes: Prevent changes to settings for Accounts & Passwords
- Cellular Data Changes: Prevent changes to cellular data settings
- Volume Limit: Prevent changes to volume settings for safe listening
- Do Not Disturb While Driving: Prevent changes to Do Not Disturb While Driving
- TV Provider: Prevent changes to TV provider settings
- Background App Activities: Prevent changes to whether or not apps run in the background."
Thank you for using Apple Support Communities.
Enjoy your day!