Add strings to an array

This initialization

char words[W][N] = {{}};

is incorrect in C. If you want to zero initialize the array then just write for example

char words[W][N] = { 0 };

In the condition of the while loop

while (scanf("%9s", test), strcmp(test, endword)) {

there is used the comma operator. Moreover you are using incorrectly the two-dimensional array test instead of a one-dimensional array

It seems you mean

char test[N];

//...

while ( scanf("%9s", test) == 1 && strcmp(test, endword) != 0 ) {

And there are used redundantly too many variables like i, j and len.

The loop could be written simpler like

char test[N];

//...

for ( ; len < W && scanf("%9s", test) == 1 && strcmp(test, endword) != 0; ++len ) 
{
    strcpy(words[len], test);
}

In this call

scanf("%d",position); 

there is a typo. You must to write

scanf("%d", &position); 

Also you should check whether the entered value of position is in the range [0, len].

For example

position = -1;

printf("\nPick the position\n");
scanf("%d", &position); 

if ( len < W && -1 < position && position <= len ) {

Also this for loop

for (i = 0; i < W-1; i++) {
  strcpy(words[i], words[i + 1]); /*Shift the words */
  words[position] = newword;
}

does not make a sense. And moreover this assignment statement

  words[position] = newword;

is invalid. Arrays do not have the assignment operator.

You need to move all strings starting from the specified position to the right. For example

for ( i = len; i != position; --i )
{
    strcpy( words[i], words[i-1] );
}

strcpy( words[position], newword );
++len;

And it seems the function convert_tolower should be called for the result array after inserting a new word. And moreover you need to pass the number of actual words in the array.

convert_tolower(words, len);

The nested loops within the function convert_tolower should look at least the following way

void convert_tolower(char matrix[][N], int n) {
  int i;
  int j;
  for (i = 0; i < n; i++) {
    for (j = 0; matrix[i][j] != '\0'; j++) {
      matrix[i][j] = tolower(( unsigned char )matrix[i][j]);
    }
  }
}

The main problem with your code was initially that you declared char *words[W][N], then tried to insert strings into this 2d array of pointers. Sparse use of organizing functions, and variables with large scopes than necessary made it hard to read. I think the best way to help you is to show you a working minimal implementation. Step 4 is not sufficiently specified. insert currently shift. It is not clear what should happen if you insert at position after empty slots, or if insert a position before empty slots and in particular if there are non-empty slots after said position.

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

#define W 5
#define N 10

void convert(size_t w, size_t n, char list[][n]) {
    for(size_t i = 0; i < w; i++) {
        for(size_t j = 0; j < n; j++) {
            list[i][j] = tolower(list[i][j]);
        }
    }
}

void insert(size_t w, size_t n, char list[][n], size_t pos, char *word) {
    // out out of bounds
    if(pos + 1 > w) return;

    // shift pos through w - 2 pos
    for(size_t i = w - 2; i >= pos; i--) {
        strcpy(list[i + 1], list[i]);
        if(!i) break;
    }
    // insert word at pos
    strcpy(list[pos], word);
}

void print(size_t w, size_t n, char list[][n]) {
    for (size_t i = 0; i < w; i++) {
        printf("%u: %s\n", i, list[i]);
    }
}

int main() {
    char words[W][N] = { "a", "BB", "c" };
    convert(W, N, words);
    insert(W, N, words, 0, "start");
    insert(W, N, words, 2, "mid");
    insert(W, N, words, 4, "end");
    insert(W, N, words, 5, "error")
    print(W, N, words);
    return 0;
}

and the output (note: "c" was shifted out as we initially had 3 elements and added 3 new words with valid positions):

0: start
1: a
2: mid
3: bb
4: end