Optimizing shared vehicle systems (bike/scooter/car/ride-sharing) is more challenging compared to traditional resource allocation settings due to the presence of \emph{complex network externalities} -- changes in the demand/supply at any location affect future supply throughout the system within short timescales. These externalities are well captured by steady-state Markovian models, which are therefore widely used to analyze such systems. However, using such models to design pricing and other control policies is computationally difficult since the resulting optimization problems are high-dimensional and non-convex. To this end, we develop a \emph{rigorous approximation framework} for shared vehicle systems, providing a unified approach for a wide range of controls (pricing, matching, rebalancing), objective functions (throughput, revenue, welfare), and system constraints (travel-times, welfare benchmarks, posted-price constraints). Our approach is based on the analysis of natural convex relaxations, and obtains as special cases existing approximate-optimal policies for limited settings, asymptotic-optimality results, and heuristic policies. The resulting guarantees are non-asymptotic and parametric, and provide operational insights into the design of real-world systems. In particular, for any shared vehicle system with $n$ stations and $m$ vehicles, our framework obtains an approximation ratio of $1+(n-1)/m$, which is particularly meaningful when $m/n$, the average number of vehicles per station, is large, as is often the case in practice.