You can create all kinds of coordinate systems where every fixed position coordinate is moving inertially and yet the coordinate system itself doesn't qualify as an "inertial frame". For example, if you have an inertial frame, and then you create a new coordinate system by doing a Newtonian Galilei transformation
on it rather than the Lorentz transformation
of SR, the resulting coordinate system wouldn't be an inertial frame despite the fact that every fixed position coordinate would be moving inertially. This coordinate system would violate the first postulate, since if you wrote down the equations for the laws of physics in terms of this coordinate system they wouldn't be the same equations that you get in inertial frames in SR. The first postulate would also be violated in a coordinate system where a light beam was at rest (since we know this is impossible in all sublight frames).
It may help to realize that part of Einstein's definition of an inertial frame was a physical
one describing how such a coordinate system could actually be constructed--this is the idea that an inertial observer constructs his rest frame using a grid of rulers at rest relative to himself, with clocks at each point along the rulers, the clocks synchronized by the Einstein synchronization convention
. Then every event is assigned coordinates using only local
readings from the ruler-marking and clock that were right next to the event as it happens (this allows you to avoid the issue of signal delays)--for example, if I see a distant explosion and I note it happened right next to the 3 light-year mark on my x-axis ruler, and the clock sitting at that mark read 15 years at the moment the explosion happened, then I would assign this event coordinate x=3 light years, t=15 years. Of course, this physical definition would make no sense for a photon, since you can't have a system of rulers and clocks moving at the speed of light (and even if you consider the limit as a system of rulers and clocks approached
c, in this limit the rulers would approach being shrunk down to zero length and the clocks would approach being totally frozen, so these limiting-case rulers and clocks would be useless for constructing a coordinate system).