Internet Explorer is a dependency of League of Legends. Many facets of League of Legends, including the store and login screen, are affected by your Internet Explorer settings. This is true even if you have a different default browser. For this reason, many common errors with the store portion of League of Legends can be resolved by restoring your Internet Explorer settings to default.
After clicking on the store button your client looks like this:
Enable TLS 1.0 and SSL 3.0 in Internet Explorer
This is a pictorial guide featuring Internet Explorer 9. If you do not have Internet Explorer 9 you can update to the latest version from Microsoft here.
1. Open Internet Explorer
2. Select Internet Options
3. Navigate to the Advanced tab
4. Scroll down in the window to the security section
5. Check the boxes to enable TLS 1.0 and SSL 3.0
6. Hit Apply
7. Restart League of Legends client
Reset your Internet Explorer settings to default
Occasionally this does not always resolve the problem. The next step is to simply reset your Internet Explorer settings to default. You can find another detailed pictorial guide to doing this here.