Would hydroponics be allowed in Israel on the seventh year, when there are additional restrictions on farming the land?
Yes they are allowed.
Due to the special status of Eretz Yisroel, generally, vegetables grown there must have ma’asros, tithes, separated. However, if the AeroGarden kit is used to grow vegetables in Eretz Yisroel, ma’asros need not be separated from the vegetables. This is because hydroponically-grown vegetables are free from the obligation to separate ma’asros.11 In addition, they may be grown during the Sabbatical year of Shmitta.12