Lyα emission is a standard tracer of starburst galaxies at high redshift. However, a number of local Lyα emitters (LAEs) are X-ray sources, suggesting a possible origin of Lyα photons other than young, hot stars, and which may be active at much later ages relative to the parent starburst. Resolved, nearby LAEs offer the opportunity to discriminate between diffuse X-ray emission arising from supernova-heated gas, high-mass X-ray binaries (HMXBs), or low-luminosity active galactic nuclei (LLAGN). We examine archival X-ray imaging from Chandra and XMM-Newton for 11 galaxies with spatially resolved Lyα imaging to determine the luminosity, morphology, and spectral hardness of the X-ray sources. The data are consistent with 9 of the 12, bright Lyα sources being driven by luminous, >10(40) erg s(-1) X-ray sources. Half of the eight Chandra sources are unresolved. The data suggest that nuclear activity, whether from LLAGN or nuclear starbursts, may play an important role in Lyα emission. Our results also suggest a significant link between Lyα emission and HMXBs, ultraluminous X-ray sources, and/or LLAGN, which would imply that Lyα may be generated over timescales 1-2 orders of magnitude longer than produced by photoionization from OB stars. This highlights a critical need to quantify the relative contributions of different sources across cosmic time, to interpret Lyα observations and the resulting properties of distant galaxies.