Is it possible to determine if a GeoJSON point is inside a GeoJSON polygon using JavasScript?

Seems like d3 has you covered: https://github.com/d3/d3-geo#geoContains

d3.geoContains(object, point)

Returns true if and only if the specified GeoJSON object contains the specified point, or false if the object does not contain the point. The point must be specified as a two-element array [longitude, latitude] in degrees. For Point and MultiPoint geometries, an exact test is used; for a Sphere, true is always returned; for other geometries, an epsilon threshold is applied.

Expanding the @thedude answer's you can also use geojson-lookup if you need to do this check more than once on a given GeoJSON.