What you are looking for is the Cut command:
editor.action.clipboardCutAction
It cuts (copies to the clipboard then deletes the line) either a selection or the entire line if nothing is selected.
I tested it with vscode 1.23.1
On windows this command is bound to Ctrl + X as well as to Shift + Delete by default.
Command works this way when Editor: Empty Selection Clipboard preference is enabled. It is enabled by default:
"editor.emptySelectionClipboard": true