How to add a sequence of numbers in a string to an integer array in c?

Here you are.

#include <stdio.h>

int main(void) 
{
    enum { N = 50 };
    int numbers[N];
    
    char *s = "12 13 14 123 1234 12345";
    
    size_t  n = 0;
    char *p = s;
    
    for ( int value, pos = 0; n < N && sscanf( p, "%d %n", &value, &pos  ) == 1; p += pos )
    {
        numbers[n++] = value;
    }
    
    for ( size_t i = 0; i < n; i++ )
    {
        printf( "%d ", numbers[i] );
    }
    
    putchar( '\n' );
    
    return 0;
}

The program output is

12 13 14 123 1234 12345

The state machine idea is good (albeit overkill; you only have three or four states after all).

You could use both this and Horner's Algorithm to parse the integer.

- set Horner accumulator to 0 and HaveNumber to 0 (false).
- loop:
    - read character
    - if digit:
       - multiply Horner by 10
       - add the value of the digit to Horner
       - set HaveNumber to 1
       - continue
    - if HaveNumber:
         push Horner onto array
         set HaveNumber to 0
    - set Horner to 0
    - if character was a 0 instead of a space:
         break
   

This does not need memory and does not necessarily use any function call (the digit check may be done verifying the ASCII value of the digit between 0x30 and 0x39)

You can use strtok to split string into substrings and then use atoi to convert that into ints.

#include <stdlib.h>
#include <string.h>

int main()
{
    char my_string[] = "12 23 234 2345";
    int my_array[50] = {0};
    
    char* token = strtok(my_string, " ");
    int i = 0;
    while (token)
    {
        my_array[i] = atoi(token);
        token = strtok(NULL, " ");
        i++;
    }
    
    return 0;
}