A Prone Creature can Dodge
Nothing in the rules specifies that a creature cannot take the Dodge action while under the effects of the Prone condition.
The Prone condition (PHB, p. 292) states:
- A prone creature's only movement is to crawl, unless it stands up and thereby ends the condition.
- The creature has disadvantage on attack rolls.
- An attack roll against the creature has advantage if the attacker is within 5 feet of the creature. Otherwise, the attack roll has disadvantage.
And the Dodge action (PHB, p. 192) states:
When you take the Dodge action, you focus entirely on avoiding attacks. Until the start of your next turn, any attack roll made against you has disadvantage if you can see the attacker, and you make Dexterity saving throws with advantage. You lose this benefit if you are incapacitated or if your speed drops to 0.
What would happen in this scenario is that any melee attackers against the prone dodging creature would not have advantage or disadvantage, while ranged attackers would have disadvantage.
There are additional rules on "Being Prone" (PHB, pp. 190-191), but it is mostly describing how to move while prone, and does not forbid the use of any actions, including the Dodge action.
(Incidentally, I'm imagining it would look like Samwise trying to dodge Shelob's stinger from TROTK film.)