The accepted answer does not work for me for nested objects for some reason. This led me to code up my own. As it's late 2019 when I write this, there are a few more options available within the language.
Update: I believe David Furlong's answer is a preferable approach to my earlier attempt, and I have riffed off that. Mine relies on support for Object.entries(...), so no Internet Explorer support.
function normalize(sortingFunction) {
  return function(key, value) {
    if (typeof value === 'object' && !Array.isArray(value)) {
      return Object
        .entries(value)
        .sort(sortingFunction || undefined)
        .reduce((acc, entry) => {
          acc[entry[0]] = entry[1];
          return acc;
        }, {});
    }
    return value;
  }
}
JSON.stringify(obj, normalize(), 2);
--
KEEPING THIS OLDER VERSION FOR HISTORICAL REFERENCE
I found that a simple, flat array of all keys in the object will work. In almost all browsers (not Edge or Internet explorer, predictably) and Node 12+ there is a fairly short solution now that Array.prototype.flatMap(...) is available. (The lodash equivalent would work too.) I have only tested in Safari, Chrome, and Firefox, but I see no reason why it wouldn't work anywhere else that supports flatMap and standard JSON.stringify(...).
function flattenEntries([key, value]) {
  return (typeof value !== 'object')
    ? [ [ key, value ] ]
    : [ [ key, value ], ...Object.entries(value).flatMap(flattenEntries) ];
}
function sortedStringify(obj, sorter, indent = 2) {
  const allEntries = Object.entries(obj).flatMap(flattenEntries);
  const sorted = allEntries.sort(sorter || undefined).map(entry => entry[0]);
  return JSON.stringify(obj, sorted, indent);
}
With this, you can stringify with no 3rd-party dependencies and even pass in your own sort algorithm that sorts on the key-value entry pairs, so you can sort by key, payload, or a combination of the two. Works for nested objects, arrays, and any mixture of plain old data types.
const obj = {
  "c": {
    "z": 4,
    "x": 3,
    "y": [
      2048,
      1999,
      {
        "x": false,
        "g": "help",
        "f": 5
      }
    ]
  },
  "a": 2,
  "b": 1
};
console.log(sortedStringify(obj, null, 2));
Prints:
{
  "a": 2,
  "b": 1,
  "c": {
    "x": 3,
    "y": [
      2048,
      1999,
      {
        "f": 5,
        "g": "help",
        "x": false
      }
    ],
    "z": 4
  }
}
If you must have compatibility with older JavaScript engines, you could use these slightly more verbose versions that emulate flatMap behavior. Client must support at least ES5, so no Internet Explorer 8 or below.
These will return the same result as above.
function flattenEntries([key, value]) {
  if (typeof value !== 'object') {
    return [ [ key, value ] ];
  }
  const nestedEntries = Object
    .entries(value)
    .map(flattenEntries)
    .reduce((acc, arr) => acc.concat(arr), []);
  nestedEntries.unshift([ key, value ]);
  return nestedEntries;
}
function sortedStringify(obj, sorter, indent = 2) {
  const sortedKeys = Object
    .entries(obj)
    .map(flattenEntries)
    .reduce((acc, arr) => acc.concat(arr), [])
    .sort(sorter || undefined)
    .map(entry => entry[0]);
  return JSON.stringify(obj, sortedKeys, indent);
}