We present new Atacama Large Millimeter/Submillimeter Array (ALMA) 850 μm continuum observations of the original Lyα Blob (LAB) in the SSA22 field at z = 3.1 (SSA22-LAB01). The ALMA map resolves the previously identified submillimeter source into three components with a total flux density of S 850 = 1.68 ± 0.06 mJy, corresponding to a star-formation rate of ~150 M ⊙ yr−1. The submillimeter sources are associated with several faint (m ≈ 27 mag) rest-frame ultraviolet sources identified in Hubble Space Telescope Imaging Spectrograph (STIS) clear filter imaging (λ ≈ 5850 Å). One of these companions is spectroscopically confirmed with the Keck Multi-Object Spectrometer For Infra-Red Exploration to lie within 20 projected kpc and 250 km s−1 of one of the ALMA components. We postulate that some of these STIS sources represent a population of low-mass star-forming satellites surrounding the central submillimeter sources, potentially contributing to their growth and activity through accretion. Using a high-resolution cosmological zoom simulation of a 1013 M ⊙ halo at z = 3, including stellar, dust, and Lyα radiative transfer, we can model the ALMA+STIS observations and demonstrate that Lyα photons escaping from the central submillimeter sources are expected to resonantly scatter in neutral hydrogen, the majority of which is predicted to be associated with halo substructure. We show how this process gives rise to extended Lyα emission with similar surface brightness and morphology to observed giant LABs.