One little inconstancy: elsewhere on that same site it specifies that it's every Republican president since Coolidge (1923-1929 has been a member. I'd normally include 1923-1929 in "since 1923". I can't find any basis for claiming Coolidge as a member. A small matter.
The best source of information I could find on the Bohemian Club was Peter Phillip's doctoral dissertation "A Relative Advantage: Sociology of the San Francisco Bohemian Club". According to that:
The Bohemian Grove has long been a political networking point for Republicans.
Dewey, Hoover, Wilkie, Eisenhower, Taft, Goldwater, Nixon, Bush, Ford, and Reagan
have all been members or guests at the Grove along with significant numbers of cabinet
members and White House officials (Van der Zee, Domhoff 1974).
This list does, indeed, include every Republican president since Coolidge (excluding GW Bush, who was elected after the thesis was written). However, that list is members and guests, not just members.
- For Hoover, Wikipedia cites the book "The Greatest Men's Party on Earth: Inside the Bohemian Grove" as stating that Hoover had joined in 1913. Phillip's thesis attests that Hoover gave lakeside chats every year from 1932 until his 1964 death.
- Phillip's thesis describes an incident of fellow-member geniality between President Eisenhower and John Howell.
- Nixon and Ford are described as members in a 1981 news report.
- Reagan: Mentioned as a guest in 1971, at which point he was the governor of California. He is described as a member in that news report as well.
- GHW Bush: Bush Sr was confirmed as a member by a White House press aide to a NY Times reporter. In Phillips' thesis:
In book shelves along the river road wall are camp scrap books, showing pictures of Mandalay Camp members going back fifty years. George Bush's picture along with Henry Kissinger and George Schultz was in the 1991 album.
- Although many conspiracy theory websites claim that George W Bush is a member, the few reputable sources I can find which mention such give no indication of how they might have discerned it - other than look at all the conspiracy theory websites that claim it.
Most other sources do not claim Coolidge as a member, and I can't confirm George W. Bush's membership (though it's certainly plausible). It does seem fairly reasonable to believe that all Republican presidents between those two were members, however.