Is there a way to attach webhooks or get events from Azure Search? Specifically we are looking for way to get notified (programmatically) when an indexer completes indexing an index.
Currently, there are no such events. However, you can implement functionality like this yourself. There are several scenarios to consider. Basically, you have two main approaches to adding content. Either define a content source and use pull or use the API to push content to the index.
The simplest scenario would be when you are using push via the API to add a single item. You could create a wrapper method that both submits your item and then queries the index until that item is found. Your wrapper method would need to either call a callback or fire an event. To support updates on an item you would need a marker on the item, like a timestamp property that indicates the time when the item was submitted to the index. Or a version number or something that allows you to distinguish the new item from the old.
A more complex scenario is when you handle batches or volumes of content. Assuming you start from scratch and your corpus is 100.000 items, you could query until the count matches 100.000 items before you fire your event. To handle updates, the best approach is to use some marker. E.g. you submit a batch of 100 updates at 2020-18-08 09:58. You could then query the index, filtering by items that are updated after the timestamp you submitted your content. Once the count from your query matches 100 you can fire your event.
You would also need to handle indexing errors or exceptions when submitting content in these scenarios.
For pull-scenarios your best option is to define a skill that adds a timestamp to items. You could then poll the index with a query, filtering by content with a timestamp after the point indexing started and then fire your event.