Hi @mushroom_posse ,

If storage account has disabled public access then you basically have 3 options:

 - setup VPN Gateway ( and use VPN to connect to your VNET)
 - setup Express Route (similar to option 1, faster but pricier)
 - manually add IPs to storage account firewall 

If your organization using firewall you can also add rules that will allow incoming traffic to your storage account from certain pool of IP addresses/FQDNs etc.