
Long-serving Kenwright, 77, defended the way the club is run in an open letter to fans last week.
“The situation is not sustainable [and] not befitting of our great club,” said the Everton Fan Advisory Board.
It added: “As Evertonians, we do not deserve it.”
Second-bottom Everton, who have been in the top-flight for 72 years, are two points adrift of safety with five games remaining.
Football: Chairman Kenwright defends Everton board
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Kenwright and the rest of the board of directors have not attended a home game since January after what the club described as “threatening correspondence” was received before a game against Southampton.
Some supporters have held peaceful protest marches before every home game since, expressing their frustration at the running of the club.
The target of their frustrations are Moshiri and the board of directors, which includes Kenwright, chief executive Denise Barrett-Baxendale, chief financial officer Grant Ingles and former Toffees striker Graeme Sharp.
Kenwright, who has been Everton chairman since 2004, says their removal will not help the team’s current on-field struggles.
“The chair of any organisation has a core responsibility for leading the board, ensuring its focus on strategic matters and direction, overseeing the company’s business, and setting high governance standards,” said the Everton Fan Advisory Board, an independent body which consults with the club’s board.
“We were therefore hugely disappointed at the chairman’s statement last Friday.
“At a critical time for our club, when we most needed leadership, understanding, accountability and a recognition of the strength of feeling and concern across large parts of our fanbase, we got the absolute opposite.
“After careful consideration, we are therefore making a call of no confidence in the current chair.
“The FAB is also calling on the majority shareholder to urgently take charge of securing stronger expertise, experience and more appropriate leadership for our club.”