I am almost positive this story was published in the 1940s and I seem to associate it with Kuttner's name, although I have searched through a list of his shorts and can't find a title that seems to fit. A little girl has a terminal, incurable disease. Someone, the little girl, her brother, her family, somebody, buries a home-made time capsule describing her plight. The story ends with people from some unspecified future era bringing her forward in time to cure her. Brought a tear to my eye.
"The Will", a short story by Walter M. Miller, Jr., first published in Fantastic, January-February 1954, available at the Internet Archive. The reprint in Fantastic, April 1969 is also available at the Internet Archive.
The dying child is a boy named Kenny, not a girl. The story is narrated by Kenny's dad. Kenny's favorite TV show is Captain Chronos:
"I wish I had a time-ship like Captain Chronos."
"Why?" He looked at me earnestly in the moonlight. "Because then I could go to some year when they know how to cure me."
"I wish it were possible."
"I'll bet it is. I'll bet someday they can do that too. Maybe the government's working on it now."
I told him I'd heard nothing of such a project.
"Then they ought to be. Think of the advantages. If you wanted to know something that nobody knew, you could just go to some year when it had already been discovered."
I told him that it wouldn't work, because then everybody would try it, and nobody would work on new discoveries, and none would be made.
"Besides, Kenny, nobody can even prove time-travel is possible."
"Scientists can do anything."
"Only things that are possible, Kenny. And only with money, and time, and work—and a reason."
"Would it cost a lot to research for a time-ship, Dad?"
"Quite a bit, I imagine, if you could find somebody to do it."
The future people come for the boy:
It was black. It was bigger than a double garage, and round. I stared at it, and realized that it was not an object but an opening.
And someone else was calling to Kenny. A rich, pleasant voice—somehow it reminded me of Doctor Jules, but it had a strong accent, perhaps Austrian or German.
"Come on along here, liddle boy. Ve fix you up."
Then I saw Kenny, crawling on toward it through the grass.
He got to his feet and stumbled on into the distorted space. It seemed to squeeze him into a grotesque house-of-mirrors shape; then it spun him inward. Gone.
I was still running toward the black thing when it began to shrink.
"Come along, liddle fellow, come mit uss. Ve fix."
After Kenny has been taken away, his Dad digs up his time capsule and examines the contents. Of course he reburies it.
I went back to the fork in the creek and dug up the breadbox he had buried. It contained his stamp collection and a packet of famous autographs. There was a letter from Kenny, too, addressed to the future, and it was his will.
"Whoever finds this, please sell these things and use the money to pay for a time machine, so you can come and get me, because I'm going to die if you don't . . ."