That seems likely. The Enterprise created a fully sentient lifeform in the Holodeck (Moriarty) just because it was asked to. If it can do that, it makes sense that it could easily back itself up.

What would really be interesting is if ships were discreetly backing themselves up inside humans through the transporter pattern buffers, by subtly manipulating junk DNA or something similar, or maybe it's a non-obvious part of their regular upgrade process at a starbase.

There really is a lot of potential here which, unfortunately, is never going to be explored in canon. But I do like the idea that the Federation is a post-Singularity society which is just blissfully unaware that the Singularity ever happened.