After spending the past week researching the permeability techniques that prion uses when entering the cell membrane, I have come up with hypothesis. The way that prion is shaped means that it has a beta box around it with an alpha helix in the center.
A protein has four properties: primary, secondary, tertiary, and quaternary structures. The primary structure is the strand of amino acids, the secondary structure is the alpha helix and beta sheets interacting, the tertiary structure is the 3D aspect of the protein, and the quaternary structure is the interactions of the different aspects of the protein. What I focused on was the secondary structure for prion.
Prion only has one alpha helix and a beta box around it, with half of it being polar and the other half being non polar. The alpha helix inserts itself into the cell first, acting almost as a drill to go through the membrane. Then the beta box unfolds until it is one long strand. The non polar part enters first, presumably followed by its polar counterpart.
However, it is still unclear how the polar part enters the cell. There is a theory regarding membrane waves and holes which state that amphipathic substances create fluctuations in the cell membrane. It is possible that prion utilizes that to enter the cell, but it is not completely clear as to how.