Unless you use a separate account for iTunes and App purchases, signing into iCloud is the sme thing. Click on this link to find out how to sign in with your Apple ID --> Sign in with your Apple ID - Apple Support "After you log in, you can use the App Store, iCloud, iMessage, Apple Music, Apple TV+, and more."
Not being able to connect has different meaning. Here are a variety of resources helping with connectivity issues:
If you can't connect to the App Store, iTunes Store, or other Apple services, read this support article --> If you can't connect to the App Store, iTunes Store, or other Apple services - Apple Support
- If none of the suggestions work, try the last one: "If you’ve tried all of the above steps and you still can’t connect, contact Apple Support."
Here's a related article about what to do if you can't connect or sign in to iCloud --> If you can't connect or sign in to iCloud - Apple Support
If you get a notification that your software version will no longer support Apple Services, some older software versions will no longer support Apple Services like the App Store, Siri, and Maps. Update your software to the latest available version to continue using these services. More information in this support article --> If you get a notification that your software version will no longer support Apple Services - Apple Support
Here's a case where using a VPN prevented sign-in --> AppStore - Apple Community
If none of the resources provide a solution, contact Apple for assistance:
For support, click here --> Choose your country or region - Official Apple Support Select your country, then a product. If you don't see one that handles your issue then keep experimenting with selections until you reach one that gets you a chat session or a telephone call and get the representative to redirect you.
or:
Contact Apple for support and service --> "See a list of Apple phone numbers around the world."
Click here --> Contact Apple for support and service - Apple Support