Had a check of some of my reference books, and unfortunately it doesn't appear that any of the Essex class ever hosted a Harrier even for trials or cross decking. Which is a pity, as so many different types of ship trialled members of the Harrier family in the 60s and 70s. Several US Carriers and Amphibs carried out trials as well as other nations Navies, here is a list. Please feel free to update:
Ship Tonnage Date of trial Aircraft
HMS Ark Royal R09 50,800 feb 1963 P1127
USS Independence CVA 62 78,000 may 1966 XV-6A Kestrel
USS Raleigh LPD 1 13,900 may 1966 XV-6A Kestrel
HMS Bulwark R08 27,300 june 1966 XV-6A Kestrel
USS La Salle LPD 3 (AGF3) 13,900 may 1969 XV-6A Kestrel
HMS Blake C99 12,100 aug 1969 Harrier GR1
ARA 25de Mayo V2 19,900 sep 1969 Harrier GR1
ITS Andrea Doria C 553 6,500 oct 1969 Harrier GR1
HMS Eagle R05 50,800 mar 1970 Harrier GR1
USS Guadalcanal LPH 7 18,300 mar 1971 AV-8A
USS Coronado LPD 11(AGF11) 16,800 mar 1971 AV-8A
RFA Green Rover A268 11,520 sep 1971 Harrier GR1
USS Guam LPH 9 18,500 jan 1972 AV-8A
INS Vikrant R11 19,500 july 1972 Harrier T2
SNS Dedalo R01 15,800 oct 1972 Harrier GR1
BNS Minas Gerais A11 19,800 oct 1973 Harrier GR1
FS Jeanne D'Arc R97 12,365 oct 1973 Harrier GR1
FS Foch R99 27,300 nov 1973 Harrier T2
USS Tripoli LPH 10 18,300 aug 1974 AV-8A
USS Inchon LPH 12 18,300 jan 1975 AV-8A
HMS Intrepid L11 12,120 jan 1975 Harrier GR1
RFA Engadine K08 9,000 jan 1975 Harrier GR1
USS Ponce LPD 15 16,800 feb 1975 AV-8A
HMS Fearless L10 12,120 jun 1975 Harrier GR1
USS Iwo Jima LPH 2 18,300 oct 1975 AV-8A
USS New Orleans LPH 11 18,300 dec 1975 AV-8A
USS Juneau LPD 10 16,800 feb 1976 AV-8A
HMS Hermes R12 28,700 apr 1976 Harrier GR1
USS Franklin D Roosevelt CV 42 64,000 jun 1976 AV-8A
USS Ogden LPD 5 16,800 aug 1976 AV-8A
USS Cleveland LPD 7 16,800 aug 1976 AV-8A
USS Trenton LPD 14 16,800 oct 1976 AV-8A
HMAS Melbourne R21 19,966 oct 1976 Harrier T2
USS Saratoga CV 60 80,000 july 1977 AV-8A
USS Shreveport LPD 12 16,800 oct 1977 AV-8A
USS Okinawa LPH 3 18,300 mar 1978 AV-8A
USS Saipan LHA 2 39,300 july 1978 AV-8A
USS Eisenhower CVN 69 91,400 aug 1978 AV-8A
USS Tarawa LHA 1 39,300 dec 1978 AV-8A
RFA Olwen A122 33,120 apr 1979 Sea Harrier FRS1
HMS Invincible R05 19,000 nov 1979 Sea Harrier FRS1
MV Atlantic Conveyer 14,946 apr 1982 Sea Harrier FRS1
HMS Illustrious R06 19,000 aug 1982 Sea Harrier FRS1
From the above list, it is interesting to note that whilst the US had a rolling programme of embarking AV-8As aboard LPHs and LHAs (logically), they were also putting an effort into expanding the range of ships that were cleared to take the Harrier from the large deck CV/CVN fleet down to the LPDs, which would only be able to operate Harriers in te VTOL mode. I suspect the latter was to enable LPDs to be used as diversion landing pads for Harriers returning from a mission if there was an accident aboard the LPH/LHA preventing landing there. In the Falklands War, two Sea Harriers used the flight decks of the LPDs Fearless and Intrepid in San Carlos Water to land and refuel whilst on CAP duty, then launched vertically and resumed patrol.