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The Advanced LIGO gravitational-wave detectors are limited by optical quantum noise in most of their detection band. To overcome this limit, squeezed vacuum states have been injected into the Advanced LIGO detectors during the third observing run (O3), leading to an increase of their detection rate by about 40% to 50%. Here we present a key element of LIGO's squeezed vacuum source: the seismic isolation platform that houses core components placed in ultra-high vacuum. This paper describes the architecture of the isolation platform as well as the active control system, tuned to minimize backscattered light that otherwise deteriorates the sensitivity of the detectors. This architecture permits fewer optical Faraday isolators in the optical path of the squeezing system, minimizing optical losses to maximize the quantum noise improvement. The system reliably operated throughout LIGO's O3 with no evidence of noise from backscattered light. The innovative architecture of this platform makes it ideal for straightforward reshaping and adaptation to other gravitational-wave detector subsystems and low-noise optical instrumentation. [ABSTRACT FROM AUTHOR]