I referred to the Airflow documentation and found that setting
auth_backends = airflow.api.aith.backends.default
in airflow.cfg only enables unauthenticated POST Requests through the experimental API. The webhook was able to post to the Airflow endpoint.
curl -X POST "https://Server/api/experimental/dags/DagInQuestion/dag_runs" \
-H "Content-Type: application/json" \
-d '{
"conf": {
"json_data": "{Data}"
}
}'
https://airflow.apache.org/docs/apache-airflow/2.7.1/security/api.html#basic-authentication