As written, there is nothing in the spell description that says the arrows are on fire, much less that they emit light.
And we should hope the arrows aren't on fire - they are magical inside our quiver:
You touch a quiver containing arrows or bolts.
A spell says when it makes something emit light, for example, holy weapon:
You imbue a weapon you touch with holy power. Until the spell ends, the weapon emits bright light in a 30-foot radius and dim light for an additional 30 feet.
This answer makes the case that fire emits light to answer a similar question about the spell create bonfire. The key difference between create bonfire and flame arrows is that create bonfire actually creates fire:
You create a bonfire.
Flame arrows imbues our arrows with a magical fire damage effect, but never says it creates any fire.
Fire damage does not have to mean open flames.
We have examples in official material of fire damage taken from sources that are not open flames. Fire damage can also mean "it's really hot". Consider the spell heat metal:
Choose a manufactured metal object, such as a metal weapon or a suit of heavy or medium metal armor, that you can see within range. You cause the object to glow red-hot. Any creature in physical contact with the object takes 2d8 fire damage when you cast the spell.
In Storm King's Thunder, it gets so hot in one room that it deals more fire damage than flame arrows:
The temperature in the room rises immediately, such that any creature that ends its turn in the room takes 5 (1d10) fire damage, or 11 (2d10) fire damage if it’s wearing metal armor.
If you want to explain the fire damage of these arrows without making them have an open flame, just make them hot. Really hot. Or make them be little microwave generators. It doesn't really matter. Your ranger will just be happy they can use their spell and Umbral Sight at the same time.
Some flames are even almost entirely invisible. Notably, ethanol fires emit very little light. See this youtube video where the inside of a plastic jug is coated in ethanol and then lit. It's quite obvious to me that there is not enough light emitted by an ethanol fire to consider the area anything other than darkness.
A DM could rule otherwise.
Sure, a DM could rule otherwise, but in the partiuclar use-case given, I would definitely recommend talking it through with the player playing the gloomstalker. As a player, I would find it quite disagreeable for one of my abilities to be nerfed by a ruling that diverges from the rules as written.