How do I make a finished hole through a lined jacket?
Expanding my comment into an answer upon further consideration:
You're right to be concerned about punching through both outer and lining material and joining them with a grommet. The lining is intended to be able to shift as you move separately from the outer jacket, and the grommet would add a small point of stress. I would also worry about the lining fraying out around the grommet.
Instead, I would unpick and resew one of the seams of the lining. Jackets with linings attached at top and bottom are often constructed by leaving the back center lining seam unsewn until the end. The jacket is turned right-side-out through that opening after being sewn to the lining inside-out, and then the lining is topstitched closed (hard to find a photo! See the last image in this bag tutorial). If that is your jacket's situation, you'll have a natural way to get inside the lining.
If not, I would still go in via unpicking a seam. A seam ripper / quick unpick is easiest, but you can be careful with some pointy scissors or a new razor blade as well. Avoid stabbing through or cutting the fabric itself (of course!). If you're getting in through a seam that is not topstitched (that is, all stitching is invisible from the outside) is helps a lot to pull the fabric on each side taut away from the seam, to whatever extent possible.
In either case, to sew it back up, I would use matching or slightly-darker thread and a small, close whipstitch. You can hide the ending thread tail between the jacket and lining by stitching down right next to your knot, into the space between the lining and jacket, and then coming back out a little ways away. Trim your thread and let it disappear inside.