Q: Can't sign in to apple account on iphone 5

Hi MikeRR5,

I'm sorry to hear you are having issues connecting to the iTunes store via Wi-Fi on your iPhone 5.

From your notes, it appears the main issue is your Wi-Fi connection as you are able to access the iTunes store with your cellular network.

The article below may be able to help you with this issue.

Click on the link below to see more details and screenshots. 

I've quoted some helpful highlights for you:

iOS: Troubleshooting Wi-Fi networks and connections

1. Be sure that you are in range of your Wi-Fi router (access point).
2. Tap Settings > Wi-Fi and turn Wi-Fi off and then on again.
   • If unable to do so, see this article.
3. Confirm that your Wi-Fi router and Cable or DSL modem are connected to power, turned on, and indicate that they are connected to the Internet.
   • If not, refer to your network administrator or ISP for assistance.
4. Restart your iOS device.
5. Tap Settings > Wi-Fi and locate the Wi-Fi network to which you are connected. Then tap the blue dot and choose Forget this Network. Try to connect to your desired Wi-Fi network again.
   • Note: You will need to enter your Wi-Fi password again if your desired network requires one.
6. Turn your Wi-Fi router off and then on again². If your ISP also provides cable or phone service, check with them before attempting this step to avoid interruption of service.
7. Update your device to the latest version of software.
8. Update your Wi-Fi router to the latest firmware².
   • For AirPort Base Stations, see this article for information about firmware updates.

If your issue is still unresolved

Follow the steps below for the issue you are experiencing.

Unable to locate a Wi-Fi network

Unable to connect to a Wi-Fi network

Wi-Fi disconnects, or signal strength is less than expected

Unable to access the Internet while connected to Wi-Fi

Unable to automatically connect to Wi-Fi

I hope this information helps ....

Have a great day!

- Judy