Calculate apsides without knowing eccentricity How would one go about determining the apsides of an orbit, knowing only the current distance between the 2 bodies, the orbital velocity, and the period? Every formula I've seen requires eccentricity, but conveniently, every formula for eccentricity requires the apsides. Surely there must be some way to find one without knowing the other?
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
$$ \mathrm{periapsis} = a ( 1 - e ) $$
$$ \mathrm{apoapsis}  = a ( 1 + e ) $$