A simple loop works well:

var str = "scissors";
var indices = [];
for(var i=0; i<str.length;i++) {
    if (str[i] === "s") indices.push(i);
}

Now, you indicate that you want 1,4,5,8. This will give you 0, 3, 4, 7 since indexes are zero-based. So you could add one:

if (str[i] === "s") indices.push(i+1);

and now it will give you your expected result.

A fiddle can be see here.

I don't think looping through the whole is terribly efficient

As far as performance goes, I don't think this is something that you need to be gravely worried about until you start hitting problems.

Here is a jsPerf test comparing various answers. In Safari 5.1, the IndexOf performs the best. In Chrome 19, the for loop is the fastest.

You can try something like the following code, from this answer:

function getIndicesOf(searchStr, str, caseSensitive) {
    var startIndex = 0, searchStrLen = searchStr.length;
    var index, indices = [];
    if (!caseSensitive) {
        str = str.toLowerCase();
        searchStr = searchStr.toLowerCase();
    }
    while ((index = str.indexOf(searchStr, startIndex)) > -1) {
        indices.push(index);
        startIndex = index + searchStrLen;
    }
    return indices;
}

You have to use brackets when accessing a field of an array. Parentheses are for calling functions.

const word = "abracadabra";
const input = ["a", "b", "c"];

for (let i = 0; i< word.length; i++) {
  if (input.includes(word[i])) {
      console.log(i + word[i]);
  }
}

You can use the RegExp#exec method several times:

var regex = /a/g;
var str = "abcdab";

var result = [];
var match;
while (match = regex.exec(str))
   result.push(match.index);

alert(result);  // => [0, 4]

Helper function:

function getMatchIndices(regex, str) {
   var result = [];
   var match;
   regex = new RegExp(regex);
   while (match = regex.exec(str))
      result.push(match.index);
   return result;
}

alert(getMatchIndices(/a/g, "abcdab"));

You can use the following code.

let str = 'Testing JavaScript, JavaScript is the Best, JavaScript is Ultimate';
function findAllIndexes(string,word){
  let result = [];
  let dif = 0;
  while(true){
    let index = string.indexOf(word);
    if(index === -1) break;
    else{
      result.push(index + dif);
      let cur = string.length;
      string = string.substring(index + word.length);
      dif += cur - string.length;
    }
  }
  return result;
}
console.log(findAllIndexes(str,"JavaScript"));

var str = "I learned to play the Ukulele in Lebanon."
var regex = /le/gi, result, indices = [];
while ( (result = regex.exec(str)) ) {
    indices.push(result.index);
}

UPDATE

I failed to spot in the original question that the search string needs to be a variable. I've written another version to deal with this case that uses indexOf, so you're back to where you started. As pointed out by Wrikken in the comments, to do this for the general case with regular expressions you would need to escape special regex characters, at which point I think the regex solution becomes more of a headache than it's worth.

function getIndicesOf(searchStr, str, caseSensitive) {
    var searchStrLen = searchStr.length;
    if (searchStrLen == 0) {
        return [];
    }
    var startIndex = 0, index, indices = [];
    if (!caseSensitive) {
        str = str.toLowerCase();
        searchStr = searchStr.toLowerCase();
    }
    while ((index = str.indexOf(searchStr, startIndex)) > -1) {
        indices.push(index);
        startIndex = index + searchStrLen;
    }
    return indices;
}

var indices = getIndicesOf("le", "I learned to play the Ukulele in Lebanon.");

document.getElementById("output").innerHTML = indices + "";

<div id="output"></div>

Try something like:

var regexp = /abc/g;
var foo = "abc1, abc2, abc3, zxy, abc4";
var match, matches = [];

while ((match = regexp.exec(foo)) != null) {
  matches.push(match.index);
}

console.log(matches);