Unlikely. And I mean that in the literal sense.
To put it simply germanium is too rare. In the Earths crust its concentration is three orders of magnitude lower than carbon. This means that odds of germanium based life on the surface of an Earth-like planet are remote.
Added issue is that germanium is not very soluble in most solvents. Some germanium compounds are but I guess the higher mass of germanium handicaps it when compared to carbon compounds.
Only real chance I see is life underground. Specifically within the core of a terrestrial planet. The cores are of presumed to be mostly of iron, nickel and siderophile elements. Germanium is one of those. Silicon and carbon are not. In there germanium could be common enough at some locations and the easiest element for complex molecules due to lack of competition. The high temperatures would remove all solubility issues.
Needless to say that is very hypothetical scenario. Our knowledge of the structure and chemistry of the core is sparse. I doubt there is a real way to evaluate the possibility of such life. From a practical view point using such life in a story might also be problematic because not only would it be entirely alien, it would be separated from familiar life by thousands of kilometers of rock hostile to either form of life.
Sound waves do travel thru that so communication might be possible. And in a story you can give the germanium sentients powerful psionic abilities and enjoy a nice game of hide and seek when human colonists try to figure out where those aliens playing mind games with them actually are.