When you asked, this wasn't possible.
With Windows Terminal 1.12 however, this is possible with the swapPane
actions. For more details, see: https://learn.microsoft.com/en-us/windows/terminal/panes#swapping-panes-preview
The syntax for these actions is:
{ "command": { "action": "swapPane", "direction": "down" } },
{ "command": { "action": "swapPane", "direction": "left" } },
{ "command": { "action": "swapPane", "direction": "right" } },
{ "command": { "action": "swapPane", "direction": "up" } },
{ "command": { "action": "swapPane", "direction": "previous" } },
{ "command": { "action": "swapPane", "direction": "previousInOrder" } },
{ "command": { "action": "swapPane", "direction": "nextInOrder" } }
But they should also be bound by default in the Command Palette, under the names "Swap Pane..."