Can I install a sub-panel with no ground to the main panel?
While the electrical code grandfathers existing installations based on the code at the time of installation, as soon as you start a new installation, you will be required to adhere to the current code.  With that in mind, regardless of the difficulty involved, you will be required to run a new line that meets all current requirements (wire type, size, burial depth, etc.) , including a dedicated ground.

If you are lucky, the existing cable is in a conduit, and assuming the new cable will fit (conduit size as required by code), you should be able to pull through a new cable with relative ease.

When in doubt, consulting a licensed electrician is always advisable.