@Matthew Spencer , Firstly, apologies you’re facing this issue. We're here to help you.
From the tag you have added here, I believe you’re levering Azure App Service WebApps and not Azure Static Web Apps.
(I'll update the tag appropriately)
If you haven’t done this already, kindly let us know if there any specific errors/runtime startup in the docker logs.
Navigate to Kudu site: https://<yourwebpp-name
>.scm.azurewebsites.net/api/logs/docker
Based on your application stack/framework, you need to add a startup file/cmd - expected values for the Startup File section.
Navigate to your WebApp on Azure Portal > Settings > Configuration> General settings > Startup Command":
See this doc for more details: Azure App Service Linux FAQ