We've been working with Azure WebJobs recently (more info: here and here) and even though the built in logging to Azure Storage is great, it is difficult to query at times.
In more recent version of the WebJobs SDK, the team exposed a TraceWriter collection via the JobHost configuration that allows consumers to write logs to a custom repository.
The snippet linked below is a naive implementation of using that collection to write the logs to SQL Server. Feel free to adapt it to fit your needs (i.e. performance concerns, async, batching, etc.)
Write Azure WebJob Logs to SQL:
Hope this helps,