Indeed. The Fermi Paradox absolutely does not rely upon FTL or even Relativistic travel speeds. A big part of Fermi's point was that even if a spreading alien civilization has to crawl from star to star at 5-10% of c, it could still spread across the galaxy in a few million years.
And while living organisms might have trouble with trips lasting in centuries, machines should be able to handle it just fine - many of our extant deep space probes have displayed unexpectedly long lifespans, attesting to the viability of such missions. A single properly designed Von Neumann Probe could easily spread to every star system in the galaxy. Certainly it is possible to imagine humanity programming just such a probe to search for alien life.