Whichever way you go to Petra it will take two or three days of your vacation Your options are:
- fly to Amman from TLV and proceed overland to Petra
- fly to Eilat from either TLV or SDV (Sde Dov airport in north Tel-Aviv) and go to Petra on a tour or on your own.
- Take a bus from either Jerusalem or Tel-Aviv to Eilat (about 5 hrs) and continue.
Tour companies that get many recommendations on the forums are "Desert Eco" (tours from Eilat) and "Abraham Tours" (two-day tours from Jerusalem or Tel-Aviv, but you only spend half a day at Petra).

HOWEVER ...... with only 7 day in total and so much to see in Israel, you should not spend two (or three) of the days traveling to and from Petra. Either extend your trip or postpone Petra to your next visit.

In Israel, you should plan AT LEAST two days in Jerusalem, a day at the Dead Sea (Massada, Ein Gedi and a short swim), a day in Tel-Aviv and the remainder in the Galilee (rent a car for a few days)..