Ensure that you're using a deployment method that includes node_modules. If you're deploying only the function's code files and package.json, without node_modules, Azure Functions runs npm install --production for you. If that's not working, you might try deploying with node_modules.
The Kudu Console or Advanced Tools (also called "Kudu services") can be accessed from the Azure portal. Navigate to your Azure Function App in the portal. Then, in the left pane, look for "Advanced Tools" under the "Development Tools" section and click "Go".
Inside Kudu, you can manually run npm install in the console (Debug console -> CMD), which might give you more information if the install is failing for some reason.