You can send Dune data to a webhook by going to any query your own, and pressing the “schedule” button to the left of “Run” in the editor. You’ll the popup below, where you can paste in a webhook url and recieve data on a scheduled interval.
A good use case for this is sending data to a Slack webhook, which you can learn to do here.
You can quickly test the output with a webhook generator like this one. Here is an example schema sent to the webhook from this Jupiter swap fees query: 
{
  "message": "Query Jupiter Fees was submitted for execution at Wed, 17 Jan 2024 04:18:49 GMT by your query schedule and it was successfully executed with a non empty result.\nYou can check its latest result here: https://dune.com/queries/3106864?utm_source=webhook&utm_campaign=alerts",
  "query_result": {
    "execution_id": "01HMAT9AB8JRV1NQR1W6NMYQK7",
    "query_id": 3106864,
    "state": "QUERY_STATE_COMPLETED",
    "submitted_at": "2024-01-17T04:18:49Z",
    "expires_at": "2024-04-16T04:20:57Z",
    "execution_started_at": "2024-01-17T04:18:49Z",
    "execution_ended_at": "2024-01-17T04:20:57Z",
    "result": {
      "data_uri": "https://api.dune.com/api/v1/execution/01HMAT9AB8JRV1NQR1W6NMYQK7/results",
      "metadata": {
        "column_names": [
          "week",
          "fee_usd"
        ],
        "result_set_bytes": 1164,
        "total_row_count": 25,
        "datapoint_count": 50,
        "pending_time_millis": 40,
        "execution_time_millis": 127845
      }
    }
  },
  "visualizations": [
    {
      "title": "Chart",
      "image_url": "https://prod-dune-media.s3.eu-west-1.amazonaws.com/screenshots/3106864/01HMAT9AB8JRV1NQR1W6NMYQK7/5222108.png"
    }
  ]
}
You’ll need to GET query the data_uri with your api key to get the results data. This is the same as passing the execution_id into get execution result.
In the case the query errors out on its scheduled run, you won’t recieve anything to the webhook.