Generate random String with a specific sign in it using JS

Just concatenate together until you got what you wanted

function stringGen(len) {
    var text = "";
    var charset = "abcdefghijklmnopqrstuvwxyz0123456789";

    for( var i=0; i < len; i++ ) {
        text += charset.charAt(Math.floor(Math.random() * charset.length));
    }

    return text;
}

stringGen(10) + "@" + stringGen(5) + "." + stringGen(3)

I also changed the initialization of the variable text to an empty string.

If you want the TLD to be without digits:

function stringGen(len, num) {
    var text = "";
    var alpha = "abcdefghijklmnopqrstuvwxyz";
    var alnum = "abcdefghijklmnopqrstuvwxyz0123456789";

    for( var i=0; i < len; i++ ) {
        if(!num)
          text += alnum.charAt(Math.floor(Math.random() * alnum.length));
        else
          text += alpha.charAt(Math.floor(Math.random() * alpha.length)); 
    }
    return text;
}

stringGen(10) + "@" + stringGen(5) + "." + stringGen(3,true)

The @ has to be after the first char and cannot be the last char. So choose a random position for it and use it if you reach this position:

function stringGen(len) {
    var text = "";
    var charset = "abcdefghijklmnopqrstuvwxyz0123456789";
    var atPos = Math.floor(Math.random() * (len - 2)) + 1;

    for( var i=0; i < len; i++ ) {
      if(i==atPos) {
        text += '@'
      } else {
        text += charset.charAt(Math.floor(Math.random() * charset.length));
      }
    }

    return text;
}

A hostname with one char or digit is allowed by a RFC ( https://serverfault.com/questions/638260/is-it-valid-for-a-hostname-to-start-with-a-digit )

Why not just write another function that uses stringGen() multiple times?

function emailAddressGen () {
  var text = stringGen(Math.ceil(Math.random() * 10 + 2)); // length of 2 to 13
  text += '@';
  text += stringGen(Math.ceil(Math.random() * 10 + 2));
  text += '.';
  text += stringGen(Math.ceil(Math.random() + 2)); // length of 2 or 3
  return text;
}

You could also refactor that Math.ceil(Math.random() * n + m) nonsense out into a random number generator helper function that takes a range (m to n) or a single max integer.

function randomNum (arg) {
  var range = [0, 0];
  if (typeof arg === 'number') {
    range[1] = arg;
  else if (typeof arg === 'object') {
    range = arg;
  } else {
    return 'Wrong argument type: must be Array or Number';
  }

  return Math.floor(Math.random() * range[1] + range[0]);
}

Try this:

////
/// @author Nikhil Saini
/// @desc Random string generator function
/// @example var key = randomString("1-a-A-$-#-*"); gives key = 6-y-Z-<-r-@
/// @invocation randomString(pattern);
/// @param pattern: A string with random character identifiers
/// @default random identifiers:-
///     1: numerical digit,
///     a: lowercase alphabet
///     A: uppercase alphabet
///     #: alphanumeric character
///     $: special character
///     *: wildcard
/// @notes:
///     1. Characters that are not acting as identifiers will be preserved
///     2. Character sets can be customised/added by setting randomStringTable.<character-identifier> to desirable string of possible characters
////

///dictionary for possible characters
var randomStringTable = {
  "1": "1234567890",
  "a": "qwertyuiopasdfghjklzxcvbnm",
  "A": "QWERTYUIOPASDFGHJKLZXCVBNM",
  "$": "~!@#$%^&*()_+`{}:\"<>?|\\[];',./-=",
  "#": "1234567890qwertyuiopasdfghjklzxcvbnmQWERTYUIOPASDFGHJKLZXCVBNM",
  "*": "1234567890qwertyuiopasdfghjklzxcvbnmQWERTYUIOPASDFGHJKLZXCVBNM~!@#$%^&*()_+`{}:\"<>?|\\[];',./-="
};

/// function for random string generation
var randomString = pattern => {
  var randomString = "";
  for (var i = 0; i < pattern.length; i++) {
    var possibility = randomStringTable[pattern[i]];
    randomString += possibility ?
      possibility.charAt(Math.floor(Math.random() * possibility.length)) :
      pattern[i];
  }
  return randomString;
};

console.log(randomString("[email protected]"));

//the function can be cascaded also
console.log(randomString("a".repeat(randomString("1"))+"@"+"a".repeat(randomString("1"))+"."+"a".repeat(randomString("1"))));

Using the code you already have, you could place the @ at a random position in the string.

String.prototype.replaceAt=function(index, character) {
    return this.substr(0, index) + character + this.substr(index+character.length);
}

var email = '873gre8g9843f98';
email.replaceAt(Math.floor(Math.random()*email.length),'@');