Azure webjobs is a feature of the azure app service and functions can also be run under app service plan. Depending on the plan utilized you can have different networking options available as mentioned here in the documentation. Below are some options which I can think of that you can use based on your requirements.
- As mentioned in this post by Ajay, due to Sandbox environment restrictions connecting to on-premises file share is not supported. You can use the REST functionality as suggested by him.
- You can use Azure Logic App to connect to your on-premises File-Share using on-premises data gateway and perform actions/triggers according to your requirements. In addition to it you can also connect your Function App to your Logic App if additional requirements are there.