Is the unit ball bounded for all metrizable topological vector spaces? If not, what would a counterexample be?

Assuming the "unit ball" to mean $\{y: d(y,0) < 1\}$, where $d$ is any metric compatible with the topology, then the answer is no.  For example, take $\mathbb R$ with the metric 
$d(x,y) = \dfrac{|x-y|}{1+|x-y|}$.  Then the unit ball consists of all of $\mathbb R$, and this is not bounded (in the topological vector space sense).
