It has been known for 40 years or more that some insects such as termites digest cellulose by synthesizing their own cellulases, whereas others have a symbiotic relationship with gut micro-organisms that perform this digestion (see Martin, Comp. Biochem. Physiol. (1983) 75A 313-324 — requires subscription — or this short popular science article).
What is the situation for silverfish — Lepisma saccharina? Today the evidence that generally determines whether an insect can digest cellulose itself is the presence of the genes for the cellulase enzymes in its genome. There is a relatively recent review of this in Annu. Rev. Entomol. (2010) 55 609–32 — again requiring subscription for access — in which the species in which these genes have been detected are listed. I know nothing about organismal biology or taxonomy, and can only summarize the species represented there for others to comment on:
This doesn’t seem to me to include Lepismatidae (which Wikipedia tells me is the family), but what do I know. (I only know the genes aren’t present in fruit flies.) However some earlier indirect experiments studying the effects of antibiotics on insect cellulase levels suggest that Lepismatidae may have their own cellulase.