The conflict between Gretchen and Squeaky in "Raymond's Run" by Toni Cade Bambara does, as aszerdi suggests, center around running. Squeaky, of course, works diligently and consistently on her running. She is fast and she always wins her races. Squeaky describes her situation this way:
I’m the fastest and that goes for Gretchen, too, who has put out the tale that she is going to win the first-place medal this year.
On the day of the race, Gretchen is waiting at the starting line, and she does appear to be Squeaky's most serious competition. Squeaky sees her "standing at the starting line, kicking her legs out like a pro," and she knows Gretchen is also running to win.
Both girls cross the finish line but it is a close finish and neither of them knows which girl won until the names are announced over the loudspeaker. Squeaky won and Gretchen came in second. Squeaky has a bit of an epiphany during the race, and she is now considering stopping her running and training her mentally challenged brother, Raymond. She thinks perhaps Gretchen might be able to help her do that, as well.
Though the two girls are certainly rivals on the track and that creates a conflict, there does not seem to be much real animosity between Squeaky and Gretchen.
In the story "Raymond's Run," Squeaky describes herself as the fastest runner in the town. She dislikes Gretchen because she has gone around the town stating that she will win first place in the races soon to take place. However, this conflict is essentially resolved at the end of the story. Gretchen proves herself to be a worthy competitor, winning Squeaky's respect. The narration ends with Squeaky thinking that Gretchen might even like to help her train Raymond.