Don't confuse buying "on line" with activating "on line". Buying on line is buying through an internet store instead of a brick and mortar store (what you are planning to do). Activating on line is the process of getting your new phone up and running.
When you receive your new phone, it will be completely blank. In order to use it, it must be activated. Do you now use an iPhone, and do you plan to "move out of" your old iPhone and "move into" your new one? If so, the startup process will guide you through the activation procedure, and all you need is a WiFi connection to facilitate the migration.
You will connect your new phone to its charger and start it up. Follow the initialization prompts, and at one point you will be asked if you want to create a new iPhone or migrate from another iPhone. Select the migrate option and you will be prompted to point the camera of one phone at the screen of the other phone to confirm the migration relationship. After a while, all of your configuration from your old phone will be replicated to your new phone (except for Face ID, wallet, and the like). After that, just move your SIM from your old phone to your new phone and you should be good to go.
Visiting your carrier's store with your old and new phones in hand will simply put this procedure in the hands of the store's clerk.
Of course, make sure that your new iPhone is not SIM locked before you order it.