Quantum computing offers unparalleled computational capabilities but faces significant challenges, including limited qubit counts, diverse hardware topologies, and dynamic noise/error rates, which hinder scalability and reliability. Distributed quantum computing, particularly chip-to-chip connections, has emerged as a solution by interconnecting multiple processors to collaboratively execute large circuits. While hardware advancements, such as IBM's Quantum Flamingo, focus on improving inter-chip fidelity, limited research addresses efficient circuit cutting and qubit mapping in distributed systems. This project introduces DisMap, a self-adaptive, hardware-aware framework for chip-to-chip distributed quantum systems. DisMap analyzes qubit noise and error rates to construct a virtual system topology, guiding circuit partitioning, and distributed qubit mapping to minimize SWAP overhead and enhance fidelity. Implemented with IBM Qiskit and compared with the state-of-the-art, DisMap achieves up to a 20.8\% improvement in fidelity and reduces SWAP overhead by as much as 80.2\%, demonstrating scalability and effectiveness in extensive evaluations on real quantum hardware topologies.