There are a couple of earlier examples of "X-ray vision" in science fiction. They are not valid answers to your question since the (not very clear) scientific basis does not seem to involve sensitivity to the X-ray portion of the electromagnetic spectrum, but I think they should be mentioned here.
1936: "The Seeing Blindness", a short story by J. Earle Wycoff in Astounding Stories, February 1936, available at the Internet Archive.
The first human trial is about to take place. The results will not be as expected:
Merton turned to the other and ran his hand nervously through his bristly
gray hair. "Here is the result of my life's work, Morris. You know its purpose?"
Morris nodded his dark head quickly. "Yes. He said you called it 'light drops'—that it enabled one to see through things. It's like the X ray, I suppose."
"Hardin should not have been so positive! But never mind. It is like the X ray, perhaps, but much more wonderful, I hope. For after this solution has taken effect in your eyes, you will be able to see through things, as you put it, but see them in their natural color! Think of being able to watch the organs of the body function naturally, with no outside influences affecting them in any way. Can't you see what it would mean in diagnosis, in operations, in setting fractures? Its possibilities are unlimited!" The doctor's blue eyes flashed with youthful enthusiasm.
1935: "Liners of Time", a novel by John Russell Fearn, first published as a 4-part serial in Amazing Stories; the "X-ray vision" was first mentioned in part 2, in Amazing Stories, June 1935, available at the Internet Archive.
Little by little, as I stood there, I began to feel the effects of the operation, so marvelously and painlessly performed. At the first my sensations were normal, then I began to sense matters take on a change. At will I could, in a sense, become as though possessed with X-ray eyesight. By a slight effort of concentration I could change my normal vision and see right through any object, in the same manner as I had done whilst on the operating table. I looked at the surgeon before me and felt a passing phase of amazement at seeing the skeleton formation of his body through his
clothes and flesh. Then I ceased to concentrate, and, to my vision, he became a solid man again.
1935: "The Man with X-Ray Eyes" a.k.a. "The Man Who Saw Everything", a short story by Edmond Hamilton, first published in Wonder Stories, November 1933, available at the Internet Archive.
Despite the title, the weird power was the result of sensitivity to ultraviolet radiation, not X-rays:
'The eyesight alteration is change enough," Dr. Homer said. "You say that you are a newspaper reporter and not a scientist. Do you realize exactly what my process involves?"
"Of course I do," David Winn answered. "I read the newspaper accounts of it
thoroughly, from the first mention of your work that appeared three months ago.
"That first article said that you. Dr. Homer, the eminent biologist of Manhattan
Foundation, believed that you could change the eyes of animals so that they could see
through stone and metal and such substances as easily as through glass.
"You proposed to do this by making the retinas of those animals' eyes sensitive to
certain ultra-violet vibrations instead of light-vibrations. They would see by these
ultra-violet radiations instead of by light, and since all inorganic matter is transparent to these particular vibrations, so would it be transparent to their eyes."
Dr. Homer nodded. "Yes, that was a fairly correct statement of my purpose in
undertaking this series of experiments. I was sure I could make animal eyes capable
of seeing through solid matter."