  We present ALMA Band 7 molecular line observations of the protostars within
the VLA 1623 system. We map C$^{17}$O (3 - 2) in the circumbinary disk around
VLA 1623A and the outflow cavity walls of the collimated outflow. We further
detect red-shifted and blue-shifted velocity gradients in the circumstellar
disks around VLA 1623B and VLA 1623W that are consistent with Keplerian
rotation. We use the radiative transfer modeling code, pdspy, and simple flared
disk models to measure stellar masses of $0.27 \pm 0.03$ M$_\odot$,
$1.9^{+0.3}_{-0.2}$ M$_\odot$, and $0.64 \pm 0.06$ M$_\odot$ for the VLA 1623A
binary, VLA 1623B, and VLA 1623W, respectively. These results represent the
strongest constraints on stellar mass for both VLA 1623B and VLA 1623W, and the
first measurement of mass for all stellar components using the same tracer and
methodology. We use these masses to discuss the relationship between the young
stellar objects (YSOs) in the VLA 1623 system. We find that VLA 1623W is
unlikely to be an ejected YSO, as has been previously proposed. While we cannot
rule out that VLA 1623W is a unrelated YSO, we propose that it is a true
companion star to the VLA 1623A/B system and that the these stars formed in
situ through turbulent fragmentation and have had only some dynamical
interactions since their inception.
