First, make sure you have the Emmet extension installed. Once you have it, Ctrl + Shift + P to go to the command pallete and search “Emmet balance Outward”.
That should select all the inner HTML and the wrapping tag. If you’re looking to only select the inner HTML, then choose Emmet Balance Inward.
Now you’ll likely want to make a shortcut to one or both of these. File > Preferences > Keyboard Shortcuts. Search for the Emmet command and associate your key combination of choice.