We use a new method based on electrostatic force microscopy (EFM) to perform quantitative measurements of the dielectric constants of individual electrospun nanofibers of poly(L-lactic acid) (PLLA), as well as composite fibers of PLLA with embedded multiwall carbon nanotubes (MWCNT-PLLA). The EFM data record the oscillation phase of an atomic force microscope (AFM) cantilever as a function of the AFM tip position. In our experiments the relative dielectric constants ϵ of the sample are measured from the EFM phase shifts vs. the tip-surface separation, according to a simple analytical model describing the tip-surface interactions. We perform a comprehensive study of how the dielectric constant depends on the fiber diameter for both electrospun PLLA and MWCNT/PLLA fiber composites. Our measurements show that EFM can distinguish between dielectric properties of PLLA fibers and fiber composites with different diameters. Dielectric constants of both PLLA and MWCNT-PLLA composite fibers decrease with increasing fiber diameter. In the limit of large fiber diameters (D > 100 nm), we measure dielectric constants in the range: ϵ = 3.4-3.8, similar to the values obtained for unoriented PLLA films: ϵ = 2.4-3.8. Moreover, the dielectric constants of the small diameter MWCNT-PLLA composites are significantly larger than the corresponding values obtained for PLLA fibers. For MWCNT-PLLA nanofiber composites of small diameters (D < 50 nm), ϵ approaches the values measured for neat MWCNT: ϵ = 12 ± 2. These results are consistent with a simple fiber structural model that shows higher polarizability of thinner fibers, and composites that contain MWCNTs. The experimental method has a high-resolution for measuring the dielectric constant of soft materials, and is simple to implement on standard atomic force microscopes. This non-invasive technique can be applied to measure the electrical properties of polymers, interphases, and polymer nanocomposites.