  Kilonovae represent an important electromagnetic counterpart for compact
binary mergers, which could become the most commonly detected gravitational
wave (GW) source. Follow-up observations, triggered by GW events, of kilonovae
are nevertheless difficult due to poor localization by GW detectors and due to
their faint near-infrared peak emission that has limited observational
capability. We show that the Near-Infrared Camera (NIRCam) on the James Webb
Space Telescope (JWST) will be able to detect kilonovae within the relevant
GW-detection range of $\sim$ 200 Mpc in short ($\lesssim$ 12-second) exposure
times for a week following the merger. Despite this sensitivity, a kilonova
search fully covering a fiducial localized area of $10$ $\mbox{deg}^2$ will not
be viable with NIRCam due to its limited field of view. However, targeted
surveys may be developed to optimize the likelihood of discovering kilonovae
efficiently within limited observing time. We estimate that a survey of $10$
$\mbox{deg}^2$ focused on galaxies within 200 Mpc would require about 13 hours,
dominated by overhead times; a survey further focused on galaxies exhibiting
high star-formation rates would require $\sim$ 5 hours. The characteristic time
may be reduced to as little as $\sim$4 hours, without compromising the
likelihood of detecting kilonovae, by surveying sky areas associated with 50%,
rather than 90%, confidence regions of 3 GW events, rather than a single event.
On detection and identification of a kilonova, a limited number of NIRCam
follow-up observations could constrain the properties of matter ejected by the
binary and the equation of state of dense nuclear matter.
