Are new galaxies able to form on the edge of the expanding universe?

It appears that you view the universe as having an edge of sorts because it appears to be expanding in three dimensions from a central point.

To the best of our knowledge, this is untrue.

A surface analogy that is more realistic would be the surface of an inflating balloon; this analogy lacks a central point and a surface edge, at least from the viewpoint of the surface itself.

As a result, our universe can be conceptualized as a three-dimensional hypersurface expanding within a four-dimensional space. It should be noted that this is only an image; as far as we are aware, our universe is not expanding into a four-dimensional space; nonetheless, it does shed light on how our universe can be expanding without having a center or an edge.

In an attempt to rephrase your question, we could ask if new galaxies are still forming. The answer appears to be that major galaxies as well as dwarf galaxies continue to form, as evidenced by observations made by NASA's Galaxy Evolution Explorer, which has seen galaxies that are much younger than our own Milky Way.

Yes, new galaxies can form at the edges of the expanding universe through the process of cosmic inflation and the gravitational pull of dark matter.