As others have mentioned, upgrading through iTunes is a workaround. I'd like to propose another solution that's worked for me.
Back your phone up to iCloud (this is probably the biggest bottleneck if your data doesn't fit into the free storage). If you have a recent backup, you can wipe your phone (Settings -> General -> Reset -> Erase all content and settings). That should clear up plenty of space on your device. Then upgrade and restore your backup once it's completed. This is how I upgraded to iOS 7 and 8 without having iTunes installed. But definitely be _sure_ you have a backup.
Back your phone up to iCloud (this is probably the biggest bottleneck if your data doesn't fit into the free storage). If you have a recent backup, you can wipe your phone (Settings -> General -> Reset -> Erase all content and settings). That should clear up plenty of space on your device. Then upgrade and restore your backup once it's completed. This is how I upgraded to iOS 7 and 8 without having iTunes installed. But definitely be _sure_ you have a backup.