As you know, metal rivets are installed in the hole then upset (hot or cold forged) to hold them in place and to hold the parts together.
Elastomeric rivets must be installed the same way. The most suitable material from which to make the rivet depends upon the total of the thicknesses of the materials to be joined (the grip of th rivet), the loading conditions (stress, vibration, cycling, etc), and the environment (chemical, thermal, etc). There are many "thermoplastic" plastics which might be considered for this application. A "thermoplastic" plastic, as opposed to a "thermosetting" plastic is one which can be remelted and reset thermally.
If you provide a few more details regarding your application, someone can probably give you more specific recommendations about what worked for them in similar situations.
Best of luck.