Plant proteins are receiving a lot of attention due to their abundance in nature, customizable properties, biodegradability, biocompatibility, and bioactivity. As a result of global sustainability concerns, the availability of novel plant protein sources is rapidly growing, while the extensively studied ones are derived from byproducts of major agro-industrial crops. Owing to their beneficial properties, a significant effort is being made to investigate plant proteins’ application in biomedicine, such as making fibrous materials for wound healing, controlled drug release, and tissue regeneration. Electrospinning technology is a versatile platform for creating nanofibrous materials fabricated from biopolymers that can be modified and functionalized for various purposes. This review focuses on recent advancements and promising directions for further research of an electrospun plant protein-based system. The article highlights examples of zein, soy, and wheat proteins to illustrate their electrospinning feasibility and biomedical potential. Similar assessments with proteins from less-represented plant sources, such as canola, pea, taro, and amaranth, are also described.