@Jack Blower Apologies for late response. Currently Annotations data is not sent to Log Analytics yet , however you can query annotations via ARM.
You need to send a GET to ARM endpoint:
Replace subscriptionId, resourceGroupName and resourceName with ones you want to query.Also replace the start and end query params value to the time duration that your annotations were created (if they are blank, server will return Invalid Request).You should also need to provide a proper Authorization token in the header.