The accumulation of thermal energy in construction elements during daytime, and its release during a colder night period is an efficient and green way to maintain a comfortable temperature range in buildings and vehicles. One approach to achieving this goal is to store thermal energy as latent heat of the phase transition using the so-called phase-change materials (PCMs). Vegetable oils came recently into focus as cheap, widely available, and environmentally friendly PCMs. In this study, we report the thermal properties of PCMs based on tamanu and coconut oils in three configurations: pure, emulsion, and encapsulated forms. We demonstrate the encapsulation of pure coconut- and tamanu-oil emulsions, and their mixtures and mixtures with commercial PCM paraffins in fiber matrices produced by a coaxial electrospinning technique. Polycaprolactone (PCL) was used as a shell, the PCM emulsion was formed by the studied oils, and sodium dodecyl sulfate (SDS) and polyvinyl alcohol (PVA) were used as emulsifiers. The addition of commercially available paraffin RT18 into a 70/30 mixture of coconut and tamanu oil, successfully encapsulated in the core of a PCL shell, demonstrated latent heats of melting and solidification of 63.8 and 57.6 kJ/kg, respectively.