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We show how Event Horizon Telescope (EHT) observations of the supermassive object at the center of M87 can constrain deviations from General Relativity (GR) in a relatively model-independent way. We focus on the class of theories whose deviations from GR modify black holes into alternative compact objects whose properties approach those of an ordinary black hole sufficiently far from the would-be event horizon. We examine this class for two reasons: ($i$) they tend to reproduce black-hole expectations for astrophysical accretion disks (and so do not undermine the evidence linking black holes to active galactic nuclei); ($ii$) they lend themselves to a robust effective-field-theory treatment that expands in powers of $\ell/r$, where $\ell$ is the fundamental length scale that sets the distance over which deviations from GR are significant and $r$ is a measure of distance from the would-be horizon. At leading order the observational impact of these types of theories arise as modifications to the transmission and reflection coefficients of modes as they approach the horizon. We show how EHT observations can constrain this reflection coefficient, assuming only that the deviations from GR are small enough to be treated perturbatively. Our preliminary analysis indicates that such reflection coefficients can already be constrained to be less than of order 10\% (corresponding to $\ell \lesssim 100 μm$), and so can rule out some benchmark cases used when seeking black-hole echoes. The precise bounds depend on the black hole spin, as well as on detailed properties of the reflection coefficient (such as its dependence on angular direction).