Motivated by the need for efficient purification methods for the recovery of valuable resources, we developed a wire-electrospun membrane adsorber without the need for post-modification. The relationship between the fiber structure, functional-group density, and performance of electrospun sulfonated poly(ether ether ketone) (sPEEK) membrane adsorbers was explored. The sulfonate groups enable selective binding of lysozyme at neutral pH through electrostatic interactions. Our results show a dynamic lysozyme adsorption capacity of 59.3 mg/g at 10% breakthrough, which is independent of the flow velocity confirming dominant convective mass transport. Membrane adsorbers with three different fiber diameters (measured by SEM) were fabricated by altering the concentration of the polymer solution. The specific surface area as measured with BET and the dynamic adsorption capacity were minimally affected by variations in fiber diameter, offering membrane adsorbers with consistent performance. To study the effect of functional-group density, membrane adsorbers from sPEEK with different sulfonation degrees (52%, 62%, and 72%) were fabricated. Despite the increased functional-group density, the dynamic adsorption capacity did not increase accordingly. However, in all presented cases, at least a monolayer coverage was obtained, demonstrating ample functional groups available within the area occupied by a lysozyme molecule. Our study showcases a ready-to-use membrane adsorber for the recovery of positively charged molecules, using lysozyme as a model protein, with potential applications in removing heavy metals, dyes, and pharmaceutical components from process streams. Furthermore, this study highlights factors, such as fiber diameter and functional-group density, for optimizing the membrane adsorber’s performance.