Well, in the 1990's when the Soviet archives were opened for examination, Tim Sebastian, a reporter for the London Times, came across a memorandum from1983 from Victor Chebrikov, the top man at the KGB, to Yuri Andropov. The memo indicated that Sen. Ted Kennedy, through his law school roommate and former Democratic Senator, John Tunney, that Kennedy would assist the Soviets in dealing with President Reagan, for which the Soviets would assist the DNC in challenging Reagan in the 1984 presidential election. Kennedy proposed he vist Moscow, “the main purpose of the meeting, according to the senator, would be to arm Soviet officials with explanations regarding problems of nuclear disarmament so they may be better prepared and more convincing during appearances in the USA.” Kennedy offered then Andropov a guarantee of interviews on American television, Kennedy would make sure certain networks would provide air time and soft-ball questions and interviews. The plan appears to be to attack Reagan on his hard-line with the Soviets painting him as a dangerous warmonger in hopes of reducing his appeal to the American voter in 1984. So...there is at least one instances where Americans may have asked for foreign powers to intervene in American elections.

/r/k1990 addressed this a bit more in depth here

Paul Kengor, The Crusader: Ronald Reagan and the Fall of Communism (Harper Collins, 2006)

Yevgenia Albats, The State Within a State (Farrar, Straus Giroux, 1994)

Reproduction of the memo HERE

In the 1892 election, a Republican operative posing as a naturalized British immigrant to the US wrote a letter to the British ambassador to the US, Lionel Sackville-West, asking which candidate would be better for British interests. Sackville-West actually fell for the trick and wrote a letter back that Democrat Grover Cleveland was better for British interests than Republican Benjamin Harrison. The GOP publicized the letter in an attempt to win over the deeply Anglophobic Irish-American vote and the letter helped Harrison reach his (very, very narrow) victory over Cleveland.