There are lots of examples.
A good one was the Whiskey Ring, which happened when Grant was president. Basically, revenue collected from liquor taxes was being stolen by government officials, agents, and liquor distributors. And in the Sanborn scandal, also under Grant, a private agent hired to collect unpaid taxes was being allowed to pocket some of the money, splitting it with the Secretary of the Treasury.
One of the biggest in US history was the Teapot Dome scandal, under President Harding. The Secretary of the Interior had convinced the US Navy to transfer him control of some oil fields, which he then leased to private companies for kickbacks. He leased them without any competitive bidding.
And of course, you had the Watergate scandal under Nixon. This involved agents and advisers of the Nixon administration performing illegal surveillance and sabotage at the Democratic National Committee's headquarters.
There are many more, but those are some of the big ones that changed the way those presidents were seen, even if the presidents themselves were not directly involved in the scandals.