Hi @James H. Robinson ,
Welcome to Microsoft Q&A Platform. Thanks for posting the query.
There is no current in-built component in Azure Data Factory to process Azure Analysis services and I would recommend you to upvote this idea in the feedback forum. All the feedback you share, is closely monitored by the Data Factory Product team and implemented in future releases.
Multiple approaches to refresh Azure Analysis databases are shared in MS Documentation using REST API, Logic Apps and Azure Automation. Using REST API and multiple components in Azure data factory primarily leveraging "Web Activity" there is a good article and corresponding code that are close references to your requirement.
Best approach to implement the above requirement depends on the flexibility of the team in designing native components or to use Logic Apps/Azure Automation.
Hope this helps! Please let us know for further details and we will be glad to assist.