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Advanced Air Mobility (AAM) operations are expected to transform air transportation while challenging current air traffic management practices. By introducing a novel market-based mechanism, we address the problem of on-demand allocation of capacity-constrained airspace to AAM vehicles with heterogeneous and private valuations. We model airspace and air infrastructure as a collection of contiguous regions (or sectors) with constraints on the number of vehicles that simultaneously enter, stay, or exit each region. Vehicles request access to airspace with trajectories spanning multiple regions at different times. We use the graph structure of our airspace model to formulate the allocation problem as a path allocation problem on a time-extended graph. To ensure that the cost information of AAM vehicles remains private, we introduce a novel mechanism that allocates each vehicle a budget of “air-credits” (an artificial currency) and anonymously charges prices for traversing the edges of the time-extended graph. We seek to compute a competitive equilibrium that ensures that: (i) capacity constraints are satisfied, (ii) a strictly positive resource price implies that the sector capacity is fully utilized, and (iii) the allocation is integral and optimal for each AAM vehicle given current prices, without requiring access to individual vehicle utilities. However, a competitive equilibrium with integral allocations may not always exist. We provide sufficient conditions for the existence and computation of a fractional-competitive equilibrium, where allocations can be fractional. Building on these theoretical insights, we propose a distributed, iterative, two-step algorithm that: (1) computes a fractional competitive equilibrium, and (2) derives an integral allocation from this equilibrium. We validate the effectiveness of our approach in allocating trajectories for the emerging urban air mobility service of drone delivery.