If you block public access to storage, always a good idea, then you still have your company's private network access. In your VNet configuration ensure your company's private network is allowed through the firewall and ensure that the storage account/container allows access from the VNet.
Here's some links to get you started.
Note that as part of the VNet firewall rules you can allow specific IPs. That is generally only done for people not on your VPN. But bear in mind that anyone not calling into your network is a security vulnerability AND their IP address will not be static. You'd need to allow a range.