Calculate apsides without knowing eccentricity
You mention this is a system in which a vessel is orbiting a planet. In that case it's fair to assume the mass of the vessel is negligible compared with the planet mass and
the planet will sit in one of the focal points of the ellipse.

You can then use Kepler's second law to find the semi-minor axis as
$$ \frac{1}{2} r v = \frac{\pi a b}{P} $$
where $r$ the current distance and $v$ the current velocity, $P$ is the period and $a$ and $b$ give the respective semi-major and semi-minor axes. 

From $b$ you find the eccentricity as
$$ b=a \sqrt{1-e^2} $$

And the apsides follow as
$$ \mathrm{periapsis} = a ( 1 - e ) $$
$$ \mathrm{apoapsis}  = a ( 1 + e ) $$