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Two-sided manufacturing-as-a-service (MaaS) marketplaces connect clients requesting manufacturing services to suppliers providing those services. Matching mechanisms i.e. allocation of clients' orders to suppliers is a key design parameter of the marketplace platform. The platform might perform an allocation to maximize its revenue or optimize for social welfare of all participants. However, individual participants might not get maximum value from their match and reject it to form matches (called blocking groups) themselves, thereby bypassing the platform. This paper considers the bipartite matching problem in MaaS marketplaces in a dynamic environment and proposes approximately stable matching solutions using mechanism design and mathematical programming approaches to limit the formation of blocking groups. Matching is based on non-strict, incomplete and interdependent preferences of participants over contracts enabling negotiations between both sides. Empirical simulations are used to test the mechanisms in a simulated 3D printing marketplace and to evaluate the impact of stability on its performance. It is found that stable matching results in small degradation in social welfare of the marketplace. However, it leads to a significantly better outcome in terms of stability of allocation. Unstable matchings introduce anarchy into marketplace with participants rejecting its allocation leading to performance poorer than stable matchings.