Flexible optics and optoelectronic devices require stretchable and compliant antireflection coatings (ARC). Conventional optical coatings, typically inorganic thin films, are brittle and crack under strain, while porous or patterned surfaces often lack environmental endurance and/or involve complex processing. Polymeric optical thin films prepared by initiated chemical vapor deposition (iCVD) comprise a promising alternative class of materials. With iCVD, multilayered, uniform thin film coatings can be synthesized conformally on the surface of a temperature-sensitive substrate near room temperature with precise compositional and thickness control. In this study, a model two-layer coating design consisting of poly(1H,1H,6H,6H-perfluorohexyl diacrylate) (pPFHDA) with a refractive index at 633 nm of n(633) = 1.426 was deposited atop poly(4-vinylpyridine) (p4VP, n(633) = 1.587). Broadband antireflection over the visible wavelength range (400-750 nm) was conferred to a transparent, flexible thermoplastic polyurethane (TPU) substrate (n(633) similar to 1.51), reducing the front-surface reflectance from similar to 4% to similar to 2%. The superior mechanical compliance of polymer ARCs over conventional inorganic coatings (MgF2, SiO2, and Al2O3) on the TPU substrate was thoroughly investigated by monitoring the evolution of film morphology and tensile fracture with applied equibiaxial strain. The polymer ARC withstood at least epsilon = 1.64% equibiaxial strain without fracture, while all inorganic coatings cracked. Through a repeated application of strain over hundreds of cycles, the antireflection by the polymer film was shown to possess excellent stability and fatigue resilience. Finally, simulations of established iCVD polymer chemistries possessing larger index contrast revealed that reflectance can be further reduced to <1% or better.