Modern supply chain operations depend on continuous customer communication, real-time shipment visibility, and rapid response to operational disruptions across increasingly complex logistics networks. Although many organizations have adopted tracking portals, transportation management systems, and customer service platforms, logistics customer support still depends heavily on manual call handling, fragmented information retrieval, and human-driven escalation workflows. These limitations create delays, increase support costs, and reduce scalability during shipment disruptions, peak seasons, and high-volume operational periods. This paper presents SupplySupport.AI, an artificial intelligence (AI) driven voice support framework designed to automate customer communication and shipment-related operational workflows in supply chain and logistics environments. The proposed framework integrates conversational AI voice agents, cloud-native orchestration, shipment tracking interfaces, secure application programming interface (API) integration, and configurable escalation logic to support natural-language customer interactions. SupplySupport.AI is intended to enable customers and operational stakeholders to request shipment status, receive delay explanations, initiate escalation workflows, and access multilingual support through voice-based conversations. The system architecture is organized around serverless cloud infrastructure, large language model (LLM) orchestration, voice synthesis, authentication controls, audit logging, monitoring, and AI guardrail mechanisms. The paper describes the motivation, architecture, workflow design, implementation model, security considerations, and a structured evaluation design for SupplySupport.AI. The primary contribution is an architecture and workflow model rather than an empirically validated system: the framework is presented at the design stage, and its anticipated operational benefits, namely reduced repetitive manual work, improved response consistency, and scalable AI-assisted shipment operations, are framed as hypotheses to be tested through prototype development, benchmarking against manual and chatbot-based support, and pilot deployment.