I think a better option would be to run Vault in an Azure Container Instance. You can find the official vault container here: https://github.com/hashicorp/docker-vault
The App Service platform execution environment differs from a local execution environment mainly due to multi-tenancy — because a single physical machine in the data center can be concurrently executing apps and services belonging to a large number of differing customers, resources are more constrained than in the case of an app running on a single machine. The sandbox mechanism mitigates the risk of service disruption due to resource contention and depletion in two ways: it (1) ensures that each app receives a minimum guarantee of resources and quality-of-service, and conversely (2) enforces limits so that an app can not disrupt other concurrently-executing apps on the same machine.
More Details on Azure App Service Sandbox: https://github.com/projectkudu/kudu/wiki/Azure-Web-App-sandbox