Vertical federated learning (FL) is a critical enabler for distributed artificial intelligence services in the emerging 6G era, as it allows for secure and efficient collaboration of machine learning among a wide range of Internet of Things devices. However, current studies of wireless FL typically consider a single task in a single-cell wireless network, ignoring the impact of inter-cell interference on learning performance. In this paper, we investigate a simultaneous transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) assisted over-the-air computation based vertical FL system in multi-cell networks, in which a STAR-RIS is deployed at the cell edge to facilitate the completion of different FL tasks in different cells. We establish the convergence of the proposed system through theoretical analysis and introduce the Pareto boundary of the optimality gaps to characterize the trade-off among cells. Based on the analysis, we then jointly design the transmit and receive beamforming as well as the STAR-RIS transmission and reflection coefficient matrices to minimize the sum of the gaps of all cells. To solve the non-convex resource allocation problem, we introduce a successive convex approximation based algorithm. Numerical experiments demonstrate that compared with conventional approaches, the proposed STAR-RIS assisted vertical FL model and the cooperative resource allocation algorithm achieve much lower mean-squared error for both uplink and downlink transmission in multi-cell wireless networks, resulting in improved learning performance for vertical FL.