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So I'm sitting and reading Spying without Spies, a book on American nuclear spying and the authors drop this tidbit.

Known about his intelligence activities during his fifteen years as a Reserve officer except for one curious incident that has recently come light. During World War I, Strauss had been secretary to Herbert who was then head of the Belgian Relief, and Strauss maintained close ties to Hoover after joining Kuhn, Loeb. In 1930, President Hoover used Strauss as a deniable link to Naval Intelligence in New York in a scenario reminiscent of the Watergate incident forty-two years later. Hoover "unofficially" asked Strauss to arrange for Naval Intelligence agents to break into the offices of an employee of the Democratic campaign manager to copy some documents allegedly harmful to Hoover's reputation. This bizarre plot was apparently carried out with Strauss's help.

Page 71 in my copy.

Is this well known? A few minutes of googling failed to find strong historical evidence for this besides this book which I'm not sure how serious it is.

What other examples of American presidents breaking the law to get elected in the 20th century are there besides Nixon and Hoover?

I found a couple of papers on Google Scholar that talk about this incident.This one mentions it on page 15, as part of a review of domestic intelligence controversies in the interwar period. It cites a book, though I don't know anything about how reliable source that book is.

This paper, which is a review essay of a book about Strauss, talks about this incident on page 107. It cites Strauss' correspondence and appointment book, as well as the private diary of Glen Howell, the intelligence agent actually designated to carry out the theft.

EDIT: It seems to me that there is little to no actual definitive evidence of this plot, as opposed to the hearsay of Howell happening to line up with incidental evidence such as Strauss's appointment book (though this all depends on how explicit Strauss' correspondence is on the issue, which isn't immediately clear). The sense I get from both of the above papers is that there was a pattern of such abuses of power throughout the interwar period, both involving Strauss in particular and broader American intelligence agencies as a whole. Thus there is little reason to dispute the evidence that actually is available, even if in a strict legal sense it might not be sufficient to convict anyone (retrospectively speaking, obviously). That's my sense based on this quick Google search, though if someone knows more about the incident or has the energy to dig through more sources on the issue they may be able to call me an idiot.