This is on the borderline between Physics and Chemistry, but I would like a Physics perspective. I am guessing that plastics are a glass-like phase, rather than a true solid.
I can't speak specifically for organic polymers, but I will try my best for polymers in general.
Every bulk polymer is made of thousands polymer chains, which is made of many "mers" (Greek for unit). Consequently we have the name polymer . For many polymers at room temperature these chains are able to rotate, and because the bonds are not 180 degrees apart this allows for the easy deformation we are familiar with (hence the colloquialism plastic). At lower temperatures the atoms do not have enough thermal energy to rotate very easily.
Bond Rotation in a Polymer Chain
In fact you are partly correct that there is a glass-like phase. The temperature that a polymer goes from a viscous fluid to an amorphous solid is called the glass transition temperature, however it should be noted that below the glass transition temperature most materials scientists will simply call the material solid.
(The definitions of solid and liquid in polymers is a little fuzzy, and most would agree the only "true" phase is the gas when all of the mers have dissociated, but at that point it's not materials science and people stop caring)
It is also important to remember that polymers become glassy in distinct crystalline regions, and the amorphous nature of the polymer is not due to molecules moving past each other like a liquid as is (eventually) the case in non-crystalline silica, but due to the rotation of the bonds.
Note that none of what I have said applies to thermo-setting polymers (epoxys) which crystallize with an increase in temperature.
Crystalline Growth of PEO and PES Seen Through Polarized Optical Microscopy
Extra info: This bond rotation also explains why trying to pull some plastic bags apart can be so easy at first and then get really difficult. At the beginning you you are simply rotating all of the chains until they are more or less aligned. After the chains become aligned you begin pulling directly against, the quite strong, carbon-carbon bonds. It is obviously much easier to simply rotate a part of a molecule, than to break covalent bonds.