Both perlite and vermiculite are open mined in various countries. Perlite is what's known as volcanic 'glass', whereas vermiculite is actually a mica like mineral. Perlite is also put through a heating process to persuade it to 'pop' in order for it to be useful in potting soil as well as other applications. Both materials have a number of other uses, notably in the construction industry; this link has a separate embedded link to information on vermiculite too https://sciencing.com/perlite-5402928.html.

Ultimately, it depends precisely what you mean by sustainable; both products are mined all the time for other uses, but the heating process for perlite could be considered non environmentally friendly. In terms of horticultural use, there is a more environmentally damaging substance contained in many potting soils, which is peat, of which there is obviously also a finite supply. This https://www.greenschoolsalliance.org/blogs/16/428 gives a little more information on ingredients used in potting soils and their environmental sustainability.