The standard approach is to use Virtual Machine data disks or VHDs that are stored as Page Blobs in Azure Storage. Take a look at Splunk in Azure Marketplace solution to easily get started with running Splunk in Azure and storing indexes in Azure Storage. This Marketplace solution encapsulates best practices and necessary steps including opening necessary ports and setting up the appropriate security groups.
You could probably do this with the Azure file service which exposes an SMB share. Performance...may not be great, or even good. That would need to be vetted out thoroughly. I would be hesitant and don't recommend this approach as a best practice.
There are no special steps required to hosting Splunk on an Azure VM. However, you must create an “endpoint" in the Azure control panel to open up communication on whichever port Splunk is running on to be able to access the Splunk Web UI remotely. See below.
Name: Splunk Web
Public Port: 80 or 8000
Private Port: 8000 [default]
Splunk ports that you might want to configure endpoints for:
9997 = Default listening port for forwarder communication.
8000 = Default Splunk web (GUI) port.
8089 = Splunk management port (also used by deployment server).