You could load a known list via this XML:

https://www.currency-iso.org/dam/downloads/lists/list_one.xml

The list was found here: https://www.currency-iso.org/en/home/tables/table-a1.html

<ISO_4217 Pblshd="2018-08-29">
  <CcyTbl>
    <CcyNtry>
      <CtryNm>
        UNITED KINGDOM OF GREAT BRITAIN AND NORTHERN IRELAND (THE)
      </CtryNm>
      <CcyNm>Pound Sterling</CcyNm>
      <Ccy>GBP</Ccy>
      <CcyNbr>826</CcyNbr>
      <CcyMnrUnts>2</CcyMnrUnts>
    </CcyNtry>
    <CcyNtry>
      <CtryNm>UNITED STATES OF AMERICA (THE)</CtryNm>
      <CcyNm>US Dollar</CcyNm>
      <Ccy>USD</Ccy>
      <CcyNbr>840</CcyNbr>
      <CcyMnrUnts>2</CcyMnrUnts>
    </CcyNtry>
  </CcyTbl>
</ISO_4217>

var xmlString = getSampleCurrencyXml();
var xmlData = (new window.DOMParser()).parseFromString(xmlString, "text/xml");
var knownCodes = [].slice.call(xmlData.querySelectorAll('Ccy')).map(n => n.textContent)

// Fetch the XML instead?
fetch('https://www.currency-iso.org/dam/downloads/lists/list_one.xml', { cache: 'default' })
  .then(response => response.text())
  .then(xmlStr => (new window.DOMParser()).parseFromString(xmlStr, "text/xml"))
  .then(data => knownCodes = data); // This may not work in the Stack Snippet

console.log(getSupportedCurrencies().map(c => c.code + '\t' + c.name).join('\n'));

function getSupportedCurrencies() {
  function $(amount, currency) {
    return Intl.NumberFormat('en-US', {
      style: 'currency',
      currency: currency,
      currencyDisplay: 'name'
    }).format(amount);
  }
  return knownCodes.reduce((currencies, cur) => {
    return (output => {
      return output.replace(/^[^ ]+ /, '') !== cur ?
        currencies.concat({
          code: cur,
          name: output.match(/(?<= ).+/)[0]
        }) :
        currencies;
    })($(0, cur).trim());
  }, []);
}

function getSampleCurrencyXml() {
  return `
    <ISO_4217 Pblshd="2018-08-29">
      <CcyTbl>
        <CcyNtry>
          <CtryNm>
            UNITED KINGDOM OF GREAT BRITAIN AND NORTHERN IRELAND (THE)
          </CtryNm>
          <CcyNm>Pound Sterling</CcyNm>
          <Ccy>GBP</Ccy>
          <CcyNbr>826</CcyNbr>
          <CcyMnrUnts>2</CcyMnrUnts>
        </CcyNtry>
        <CcyNtry>
          <CtryNm>UNITED STATES OF AMERICA (THE)</CtryNm>
          <CcyNm>US Dollar</CcyNm>
          <Ccy>USD</Ccy>
          <CcyNbr>840</CcyNbr>
          <CcyMnrUnts>2</CcyMnrUnts>
        </CcyNtry>
      </CcyTbl>
    </ISO_4217>
  `;
}

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If you want to generate the codes still, you can use a product iterable.

The following is based on Python's itertools.product function.

let ary = product('ABCDEFGHIJKLMNOPQRSTUVWXYZ'.split(''), 3).map(a => a.join(''));

function product(iterables, repeat) {
  var argv = Array.prototype.slice.call(arguments), argc = argv.length;
  if (argc === 2 && !isNaN(argv[argc - 1])) {
    var copies = [];
    for (var i = 0; i < argv[argc - 1]; i++) { copies.push(argv[0].slice()); }
    argv = copies;
  }
  return argv.reduce((accumulator, value) => {
    var tmp = [];
    accumulator.forEach(a0 => value.forEach(a1 => tmp.push(a0.concat(a1))));
    return tmp;
  }, [[]]);
}

Demo

console.log(getSupportedCurrencies().map(c => c.code + '\t' + c.name).join('\n'));

function getSupportedCurrencies() {
  function $(amount, currency) {
    return Intl.NumberFormat('en-US', {
      style: 'currency',
      currency: currency,
      currencyDisplay: 'name'
    }).format(amount);
  }
  let ary = product('ABCDEFGHIJKLMNOPQRSTUVWXYZ'.split(''), 3).map(a => a.join(''));
  return ary.reduce((currencies, cur) => {
    return (output => {
      return output.replace(/^[^ ]+ /, '') !== cur
        ? currencies.concat({ code : cur, name : output.match(/(?<= ).+/)[0] })
        : currencies;
    })($(0, cur).trim());
  }, []);
}

function product(iterables, repeat) {
  var argv = Array.prototype.slice.call(arguments), argc = argv.length;
  if (argc === 2 && !isNaN(argv[argc - 1])) {
    var copies = [];
    for (var i = 0; i < argv[argc - 1]; i++) { copies.push(argv[0].slice()); }
    argv = copies;
  }
  return argv.reduce((accumulator, value) => {
    var tmp = [];
    accumulator.forEach(a0 => value.forEach(a1 => tmp.push(a0.concat(a1))));
    return tmp;
  }, [[]]);
}

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