Are there expanding-universe cosmological models that do not include inflation?
Variable speed of light (VSL) theories (for instance Petit (1988) and Moffat (1993)) have been introduced in the past to solve the horizon problem you mentioned, i.e. that the observable universe appears to be isotorpic and homogeneous. If certain universal "constants" (including $c$, the speed of light) are allowed to change, then it is possible for interactions between extremely distant sections of the universe to occur, solving the horizon problem. These modifications still fit in with general relativity and the idea of an expanding universe.

VSL theories, though, have not gained much traction, and inflation is still the prevailing theory in the scientific community today. It solves the horizon problem, the flatness problem (dealing with how fine-tuned the initial density of the universe appears to have been), and the magnetic monopole problem (i.e. why we don't see magnetic monopoles). Inflation isn't the only possible solution to these problems, but it works quite well.