In which animated cartoons did the Coyote even come close to catching the Road Runner? I know that he caught him once, but he didn’t know what to do afterwards.
Wile E Coyote finally catches his prey in a segment titled "Soup or Sonic" which is part of a TV special produced in 1980. This is apparently the only time in "official" productions.
And the gag is, as you recall, the Coyote has no idea what to do with the Road Runner and is asking the audience for advice/instruction.
According to this video there are an additonal two cartoons besides for the accepted answer where Wile E Coyote catches the Road Runner (I am not sure how to embed it): "Hopalong Casualty" (Chuck Jones, 1960) and "The Solid Tin Coyote" (Rudy Larriva, 1966).
I believe it depends how one defines "catch". In these two cartoons, Wile E. Coyote does catch the road runner, but the road runner manages to escape. However, OP asked for cartoons where Wile E Coyote "came close to catching" and these two cartoons definitely fit the bill.