In looking at your image, it seems that like your local machine is failing to reach npmjs.org. It doesn't on the surface seem Azure deployment related but one suggestion I would try is zip the bot up and deploy manually. Unfortunately, I don't think we have any tags related to on-premises network troubleshooting but I'll try to help the best I can.
Does it build and run fine locally and is able to get the dependent packages? Does a tracert
to registry.npmjs.org successfully run when running behind the client's firewall? Do these errors persist when you're not on the client's network?