It is true that space is expanding[citation needed] and due to this expansion, over time, distant objects will become unobservable. Eventually distant galaxies will be beyond our Particle Horizon (the maximum distance a particle of light could be and still be observed by us). In the far flung future, if we're still around, the observable universe will essentially be the closest galaxies to us.
However, that effect is still very, very minimal in our young universe. Galaxies in our universe did not form immediately. Nor has our universe been expanding at the same rate for its entire existence. Those two concepts combined essentially mean that even the earliest galaxies are still within our particle horizon.
As a specific example, Wikpedia's List of the most distant astronomical objects show's that the most distant objects we have found are at redshifts in the range of 6 to 11. This represents an age of a few hundreds of millions of years after the Big Bang.
A relevant source to this discussion is Loeb (2001) which states:
objects with the current redshift z from 5 to 10 will remain observable for no more than 4–6 billion years
While the early galaxies are far off and on the way out of our particle horizon, we still have a long time to go (on a human scale) before we won't be able to see these objects "shortly" after the Big Bang. Just to give you a sense of the time scales involved in this process, Loeb (2001) states
Within $\lesssim 10^{11}$ years, we will be able to see only those galaxies that are
gravitationally bound to the Local Group of galaxies, including the Virgo cluster and possibly some parts of the local supercluster
This means it'll be a few billion years before the most distant objects are no longer observable and a hundred billion years before almost everything is no longer visible.
What's more, even once they're outside our event horizon (meaning no newly emitted light will reach us) it still takes billions of years for the already emitted light to finally reach us meaning we'll still see the objects long after they're "unobservable" Loeb (2001) shows this concept in some of the equations and plots.