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Stuart Crawford, a former lieutenant colonel in the Royal Tank Regiment and member of East Lothian Lib Dem executive, also called for a change in the party's stance on independence.
His comments come after Kit Fraser quit as convener of East Lothian Lib Dems in protest at Mr Rennie's "bizarre" photo stunts during the election.
The party leader was pictured taking karate lessons on the Meadows, visiting the penguins and Edinburgh Zoo and driving a supercar at Ingliston racing circuit.
Mr Crawford said the Lib Dems' poor results at the Holyrood elections demanded a change of leader.
"Willie Rennie nice chap though he is, has presided over a decline in party success over his ten years of leadership. I think he needs a break, and another needs to step up to the plate. Perhaps it’s time for Alex Cole-Hamilton to test his mettle in the role, or does the party leader really need to be an elected member at all?"
Writing on the Think Scotland website, he lamented the Lib Dems' reduction to four MSPs, losing the major party status which gave them a guaranteed slot at First Minister's Questions and a seat on parliament's business bureau.
"In essence we no longer have a party operating at Holyrood, just four party members with jobs as MSPs."
He dismissed the party's election slogan “Put Recovery First” as "motherhood and apple pie" which no-one could disagree with.
He claimed the Lib Dems' position on independence had been too close to the "robustly anti-referendum Conservatives and punished by the electorate accordingly".
And he said East Lothian Lib Dems had proposed a "more nuanced" approach – "we should change from being anti-referendum to a position of neutrality on the referendum per se, whilst retaining the option to campaign against independence and for the Union as the wider party membership chose".
A Scottish Lib Dem spokesman said: "Willie Rennie ran an energetic campaign to make sure the new parliament is focused on recovery, not independence."