You can send Dune data to a webhook by going to any query you own, and pressing the “schedule” button to the left of “Run” in the editor. You’ll see the popup below, where you can paste in a webhook url and receive 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 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 receive anything to the webhook.