No time to check but:

• scanf("%4d", &S) or something
• you have if, else if, but no simple else in the end
• add \n to printf

Last but not least: move the code after scanf to its own function that takes 1 number XXXX or 2 numbers (XX and YY), and call it in a loop for (int i = 0; i < 9999; ++i) and print the results like: "p1, p2, result", that way you'll quickly check which value is invalid or not.

If input is 0 then the program prints nothing instead of NA

What happens in your program when scanf stores -1 as the value, like if it hits EOL before finding matching input?

Maybe you covered this and I missed it.

This would be easier to understand if you defined a function like:

char IsMonth(int value){
    return value > 0 && value <= 12;
}

Then try to rewrite your program using that inside of the if statements. For example

if(IsMonth(p1)){
    ...
}

If you do that, I'm pretty sure you'll find the problem. As a hint, notice that IsMonth is a boolean value, since a number can only either be a month (true) or not a month (false). I haven't debugged it, but I can tell there's a problem just by the number of printf statements you currently have.

Maybe something like this

#include <stdio.h>

// reads next 4 chars. doesn't check what's beyond that.
int get_pair() {
  int h = getchar() - 48;
  int l = getchar() - 48;

  return h * 10 + l;
}

int main(){
  int p0 = get_pair();
  int p1 = get_pair();
  if (p0 < 0 || p1 < 0 || p0 > 100 || p1 > 100) {
   // not 4 digi seq, return with failure if that's a requirement 
  }

  if ((p0 == 0 || p0 > 12) && (p1 >= 1 && p1 <= 12)) {
    printf("YYMM");
  } else if ((p1 == 0 || p1 > 12) && (p0 >= 1 && p0 <= 12)) {
    printf("MMYY");
  } else if ((p0 >= 1 && p0 <= 12) && (p1 >= 1 && p1 <= 12)) {
    printf("AMBIGUOUS");
  } else {
    printf("NA");
  }
  return 0;
}

or if you want to optimize

#include <stdio.h>
#include <stdint.h>

// reads next 4 chars. doesn't check what's beyond that.
int get_pair() {
  int h = getchar() - 48;
  int l = getchar() - 48;

  return h * 10 + l;
}

uint8_t props (int p) {
  if (p >= 1 && p <= 12) {
    return 0b10;
  } else if (p < 0 || p >= 100) {
    return 0b11;
  } else {
    return 0b00;
  }
}

int main(){
  int p0 = get_pair();
  int p1 = get_pair();

  switch (props(p0) | (props(p1) << 2)) {
    case 0b1010: printf("AMBIGUOUS"); break;
    case 0b1000: printf("YYMM"); break;
    case 0b0010: printf("MMYY"); break;
    default: printf("NA");
  }
  return 0;
}

If the input is anything of the form 00XX, then p1 == 0. This will fail for p1 != 0 as well as p1 >= 13, meaning neither of the top-level ifs will trigger.