Your entire shipping team should agree to this change since it will affect all users in your ShipStation account. That aside, the process is really simple.
Head to your Account Settings (the wrench icon in the upper right).
Next, select Shipping and then Carriers & Fulfillment from the sidebar on the left.
Once there, click the Account Settings link for the shipping account that you'd like to promote to Primary status.
This can vary from carrier to carrier, but look for an option along the lines of "Make this my primary account."
Select that option and confirm any changes as needed. This also 'demotes' whichever account used to be your Primary account.
All orders will now default to the primary account unless assigned a different account.
Orders assigned to use specific shipping accounts won't have their selections change if you promote to or demote from Primary status. In these cases, the (Primary) indicator will simply show up or go away as appropriate.