Atopic dermatitis (eczema) is a widespread disorder, with researchers constantly looking for more efficacious treatments. Natural oils are reported to be an effective therapy for dry skin, and medical textiles can be used as an alternative or supporting therapy. In this study, fibrous membranes from poly(vinyl butyral-co-vinyl alcohol-co-vinyl acetate) (PVB) with low and high molecular weights were manufactured to obtain nano- and micrometer fibers via electrospinning for the designed patches used as oil carriers for atopic skin treatment. The biocompatibility of PVB patches was analyzed using proliferation tests and scanning electron microscopy (SEM), which combined with a focused ion beam (FIB) allowed for the 3D visualization of patches. The oil spreading tests with evening primrose, black cumin seed, and borage were verified with cryo-SEM, which showed the advantage nanofibers have over microfibers as carriers for low-viscosity oils. The skin tests expressed the usability and the enhanced oil delivery performance for electrospun patches. We demonstrate that through the material nano- and microstructure, commercially available polymers such as PVB have great potential to be deployed as a biomaterial in medical applications, such as topical treatments for chronic skin conditions.