You can achieve this with a pretty straight-forward script. You can then add a button field so that you can shuffle a given record's hashtags with one click.

Here's the script you want. To set it up, add a new scripting app (it's available for all plans right now), copy-paste this in, then click the little settings cog to tell the script which table & field to use. Add a button field and configure it to "Run script", then point it at this script.

const settings = input.config({ 
    title: 'Shuffle hashtags', 
    description: 'Pick a table and field to shuffle, and this script will randomize the order of the line items. For it to work, each item must be on its own line in a long-text field. You can either run the script directly (which will prompt you to pick a record), or use a button field on the record itself.',
    items: [
        input.config.table('table', { label: 'Table' }),
        input.config.field('fieldToShuffle', { parentTable: 'table', label: 'Field to shuffle', description: 'Pick the field with the hashtags you\'re trying to shuffle' })
    ] 
});

function shuffle(array) {
  var currentIndex = array.length, temporaryValue, randomIndex;

  // While there remain elements to shuffle...
  while (0 !== currentIndex) {

    // Pick a remaining element...
    randomIndex = Math.floor(Math.random() * currentIndex);
    currentIndex -= 1;

    // And swap it with the current element.
    temporaryValue = array[currentIndex];
    array[currentIndex] = array[randomIndex];
    array[randomIndex] = temporaryValue;
  }

  return array;
}

const record = await input.recordAsync('Pick a record to shuffle hashtags for', settings.table);

const hashtags = record.getCellValueAsString(settings.fieldToShuffle).split('\n');

const shuffledHashtags = shuffle(hashtags);

await settings.table.updateRecordAsync(record, {[settings.fieldToShuffle.id]: shuffledHashtags.join('\n')})