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As mentioned in another answer, we need to use unsigned char with tolower():

int ci_strcmp(const unsigned char *s1, const unsigned char *s2)
{
    for (;;) {
        int c1 = tolower(*s1++);
        int c2 = tolower(*s2++);
        int result = (c1 > c2) - (c1 < c2);
        if (result || !c1) return result;
    }
}

/* Adapter for qsort */
int scmp(const void *p1, const void *p2)
{
    const unsigned char *const *sp1 = p1;
    const unsigned char *const *sp2 = p2;
    return ci_strcmp(*sp1, *sp2);
}

• We missed a few necessary includes for the comparison function:

  #include <ctype.h>
  #include <stdint.h>
  #include <string.h>
  

  And for the test program:

  #include <stdio.h>
  #include <stdlib.h>
  

• We're assigning char* pointers using string literals. Whilst C allows this for historical reasions, it's dangerous, and should be avoided (because assignment through such pointers is UB, and the compiler can't spot that for you). It's easy to fix that:

  const char* strings[4] = {"Onus", "deacon", "Alex", "zebra"};
  

• We missed a few necessary includes for the comparison function:

  #include <ctype.h>
  #include <stdint.h>
  #include <string.h>
  

  And for the test program:

  #include <stdio.h>
  #include <stdlib.h>
  

• We're assigning char* pointers using string literals. Whilst C allows this for historical reasons, it's dangerous, and should be avoided (because assignment through such pointers is UB, and the compiler can't spot that for you). It's easy to fix that:

  const char* strings[4] = {"Onus", "deacon", "Alex", "zebra"};
  

And one that GCC can't spot:

qsort(strings, 4, 8, scmp);
//               ^^^

We can't assume that a const char* is a particular size on the target platform. Whilst this value may be correct where you are, it's not true everywhere. Luckily, C provides the sizeof operator to help:

qsort(strings, 4, sizeof strings[0], scmp);

We might consider including some non-alphabetic characters, too ("should "A-team" sort before or after "abacus"?).

Looking in detail at the comparison function, it does a lot of extra work to make a copy of each input string (we have to traverse each string up to three times - once to find its length, once to copy and convert case, and once to compare.

We might consider including some non-alphabetic characters, too ("should "A-team" sort before or after "abacus"?). And we'll want some tests of equal and almost-equal strings; not just ones that differ in the first character.

Looking in detail at the comparison function, it does a lot of extra work to make a copy of each input string (we have to traverse each string up to three times - once to find its length, once to copy and convert case, and once to compare).