Can you upgrade storage without getting a new phone?

You cannot upgrade the physical space on your iOS device without buying a new phone. However, you posted this in the iCloud area and that can be upgraded regardless, but won't affect the free space on your phone. -AJ

You posted in the iCloud forum, so I assume you're asking about iCloud storage. Purchasing iCloud storage will not add to your device's storage. The storage drive in your iPhone is a physical drive inside the phone, buying digital storage cannot expand it. If you're looking to get more device storage, you would need to purchase a newer, higher capacity phone. If you really are trying to get more iCloud storage, just go to Settings > iCloud > Storage > Change Storage Plan.