It’s up to the DM; however, it is likely that the Shadowfell is not pitch black all the time in 5e.
As you noted, the description of the Shadowfell includes the following:
It is a place of darkness that hates the light, where the sky is a
black vault with neither sun nor stars. (DMG pg 51)
For the purposes of this discussion, Domains of Dread and other demiplanes within the Shadowfell will be ignored, as each will presumably have its own characteristics - for example, Curse of Strahd indicates Barovia has bright light during the day that is not considered sunlight (CoS pg 24).
Also for purposes of this discussion, “pitch black” is presumed to be synonymous with darkness as defined in the rules on vision and light (PHB pg 183).
To support this answer, it needs to be shown that illumination in the Shadowfell is both a) possible and b) likely.
A) Illumination is Possible
While there is neither sun nor stars in the sky of the Shadowfell, there may be other ambient light. We have an example of another plane of existence with neither sun nor stars that still has illumination.
The description of Avernus, one of the Outer Planes, indicates the following:
The hellscape of Avernus sits under hideous clouds that obscure the
vault of the sky, from which the occasional meteor streaks before
crashing to the ground. Ambient light continually swells up from just
below the horizon as though lit by nine setting suns, yet no actual
celestial bodies fill the sky – no sun, moons, or stars. This constant
twilight makes it difficult for the denizens of Avernus to track the
passage of time. (DiA pg 76)
Accordingly, a DM could decide that - similar to Avernus - the Shadowfell has ambient light emanating from just below the horizon, if not from the vault of the sky itself. Rather than (presumably) the literal fires of the Hells in Avernus, it could be something more appropriate for the Shadowfell. One choice could be moonlight.
While the Shadowfell has no sun nor stars, unlike Avernus the description of the Shadowfell does not explicitly rule out moonlight. So it may be possible that moonlight emanating from just below the horizon is providing sufficient illumination that it is not pitch black all the time.
Recall that the Shadowfell is an Echo Plane that in places will reflect a world such as Toril. This could possibly include its moon, Selune, and her consorts, the Tears of Selune. According to the description of the goddess Selune:
The Tears of Selune, the cluster of starry lights that follow the moon
around the sky, are thought to be brought about by the goddess’s joy,
sorrow, or both.” (SCAG pg 36)
A DM could decide that Selune and the Tears of Selune are not stars, but do give off their own light (as opposed to reflected sunlight) or alternatively reflect sunlight from far beneath the horizon or some other magical effect, whatever best suits their campaign.
Whereas on the Prime Material plane such "moonlight" would still not really be enough to create even dim light conditions (refer to the rules in PHB on vision and light referenced above), a DM could decide that the features of a plane such as the Shadowfell make it possible that such "moonlight" or other ambient light does create enough light for at least dim light conditions. Similar to fires not being able to light up an entire landscape but (apparently) in Avernus they provide sufficient illumination for dim light across the entire plane. (Note that twilight is defined in the rules about vision and light as providing dim light conditions (PHB pg 183).
Accordingly, there is an argument that illumination in the Shadowfell is possible, despite the lack of sun and stars.
B) Illumination is Likely
Common sense would suggest that there is at least dim light during some of the time in the Shadowfell. It is inhabited by Vistani, some of whom are likely to be human and thus without darkvision. However, more than that . . . remember . . . the description of the Raven Queen (emphasis mine):
The Fortress of Memories
The fortress is a mournful place, filled with incessant echoes of the
past. Flocks of ravens that act as her eyes and ears darken the skies
around it when they emerge from within, bearing her cryptic messages
and omens far and wide across the multiverse. (MToF pg 59)
This seems to strongly imply that the Shadowfell is not already pitch black all the time and that at least some of the time there is ambient light in the Shadowfell, as skies that were already pitch black are unlikely to be “darkened” by such flocks.
Assuming a DM takes this approach, they may want to consider the brightness of any light in the Shadowfell, and whether there are fluctuations between “days” and “nights”. Both possibilities appear reasonable:
Constant dim light, similar to the twilight of Avernus where there are no days or nights – could fit in with a theme of being very dreary in the Shadowfell, not even having the changing of day to night, or
“Days” and “Nights” where it fluctuates from either dim or bright light during the “day”, to dim light or darkness at “twilight” and “dawn”, to darkness at “night” (somewhat similar to fluctuations of the constant sun in the Elemental Plane of Fire). (DMG pg 55)