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A recent line of research has established a novel desideratum for designing approximately-revenue-optimal multi-item mechanisms, namely the buy-many constraint. Under this constraint, prices for different allocations made by the mechanism must be subadditive, implying that the price of a bundle cannot exceed the sum of prices of individual items it contains. This natural constraint has enabled several positive results in multi-item mechanism design bypassing well-established impossibility results. Our work addresses the main open question from this literature of extending the buy-many constraint to multiple buyer settings and developing an approximation. We propose a new revenue benchmark for multi-buyer mechanisms via an ex-ante relaxation that captures several different ways of extending the buy-many constraint to the multi-buyer setting. Our main result is that a simple sequential item pricing mechanism with buyer-specific prices can achieve an $O(\log m)$ approximation to this revenue benchmark when all buyers have unit-demand or additive preferences over m items. This is the best possible as it directly matches the previous results for the single-buyer setting where no simple mechanism can obtain a better approximation. From a technical viewpoint we make two novel contributions. First, we develop a supply-constrained version of buy-many approximation for a single buyer. Second, we develop a multi-dimensional online contention resolution scheme for unit-demand buyers that may be of independent interest in mechanism design.