It sounds like a silly question, but it caused way more discussion the other night in my Adventurer's league game than I'd care to admit...
In 5e, the prone condition is described as:
A prone creature’s only movement option is to crawl, unless it stands up and thereby ends the condition.
A snake's only option is to crawl, and isn't capable of standing, as it has no limbs.
Is a snake (or snake-like creature) considered constantly prone?
If not, can such a creature be knocked prone?
Prone is a condition (PHB, 290). And the only way to avoid that condition is to have some form of immunity or something else mechanically that prevents a creature from getting the condition.
For creatures, the statblock lists Condition immunities (if there are any). Spells, items, etc. will also state if there are any conditions that they prevent/mitigate.
No snake statblock lists immunity to the prone condition.
With no immunity to the condition Prone, they are susceptible to that condition and its effects.
Jeremy Crawford has confirmed this as well:
If a monster is immune to the prone condition, that fact is noted in its stat block. Even legless creatures can be knocked over, destabilized, and the like (I've watched an upended snake flop around in real life—shudder).
Traditionally, we imagine prone as lying on one's stomach (and the associated 5e mechanic of crawling) - which is synonymous with a snake's normal movement of crawling. However, given that there is no condition immunity to snakes for Prone, then we have to think of it someway else. Maybe the snake is disoriented, or on its back, or in some position that leaves it with the same mechanical disadvantage as a bipedal creature on its stomach.
* Tastes like chicken
As mentioned in other answers, the Monster Manual generally states when a creature is immune to a certain condition, such as prone.
Therefore, since this is not the case for the snake, it isn't considered to be immune to the prone condition.
Also, to address the first part of your question - no, a snake is not considered prone all the time, otherwise this would be explicitly stated, judging by my experience of 5e's approach to rules.
I get your confusion as to how a "prone" snake would be any different from the normal body position of a snake, so I did a little google research (basically, I just typed in "d&d snake prone") and stumbled upon this reddit post, which targets pretty much the same issue you had. The first answer states the following (emphasis mine):
I would argue the the intention of 'prone' is that the creature is in an awkward position in which it is not accustomed to moving or defending itself. A snake lives most of its life flat on the ground, so that is its normal state, and I would not impose the usual prone status.
However, if the snake were to be flipped over, this would be unusual for the snake, and it would need to take time to roll itself over. Prone would apply in that case.
So, for the purpose of imagining a "prone" snake, I suggest treating a snake with the prone condition as described by DigitalSoul247 in the quoted paragraph above.
Monster Manual presentes immunity to prone as a condition immunity. Creatures that are immune to being prone have this trait: examples include swarms (pages 337-339) and oozes (page 241-243).
I didn't do a comprehensive look-up on snakes and snake-like creatures, but at least the Poisonous Snake on page 334 lacks any condition immunities - meaning it is likewise suspectible to being prone.
As the Monster Manual errata doesn't change the situation, snakes being proned is indeed in line with the current rules.
The snakes prone position is going to indicate they are laid out and not in a position to coil and strike easily. If a snake is in a forced prone position that means they would not have the ability to easily get into a striking position. If a snake is knocked prone then they would need to spend half of their movement to get into a striking position the next round, or suffer the penalties of remaining prone including reduced movement.
As a house rule I would treat any moving snake as prone, and only require the movement penalty as the result of their movement being intercepted or using double move for full distance. A snake in normal unchallenged movement(IE not moving out of challenge range and provoking an successful attack of opportunity). Would not need to spend the movement to coil, and I would treat any snake not so challenged to have finished their round in an upright coiled position. This is because of the way the game mechanics work not because of a reality situation.
The regular, "un-prone" postures for a snake are coiled, sidewinding or "zig-zagged". In these postures, the snake can move or attack (or possibly even evade) efficiently and fast. The head (and, for a rattlesnake, also the tail) may be elevated.
The prone posture would be a position in which the snake's ability to move is impeded. A tactical (voluntary) prone position might be assumed in a trench or along the foot of a wall, where the snake can not be seen as easily, compared to the regular coiled, sidewiding or "zig-zagged" wavy posture. An involuntary prone position might be caused by someone kicking or throwing a snake into a posture or location from where it cannot easily assume an efficient movement or attack posture. A "straightened" snake will need to re-coil first before it can to move or attack. "Fun moves" against snakes could be:
The other answers already provide details on how this correlates to the game rules.
5e defines prone, and it aligns with the definition of prone in English. 5e is written in English.
In both 3e and 4e, you are supposed to treat keywords as magic tokens and mostly ignore what they mean in English. In 4e they even gave this advice explicitly in the rules text.
This is not how 5e is written. 5e is, first and foremost, an English text.
From the 2014 PHB:
Combatants often find themselves lying on the ground, either because they are knocked down or because they throw themselves down. In the game, they are prone.
Snakes cannot become "more prone" - they are already lying on the ground. Making them "more prone" is nonsense. Applying the prone condition to a creature that is already lying on the ground should do nothing. Well, if the "snake" had wings, I would probably make it fall, as knocking things prone in 5e also knocks (non-hovering) creatures out of the sky.
It is consistent to invent a definition of "Prone" that isn't the English word "prone", but there is no huge balance effect here, but there isn't a need to do so.
Use common sense, and then double check it to ensure it doesn't break the game part of the game. By making it impossible to trip snakes, are you making a PC's entire build worthless? Are you breaking an entire game subsystem? Does this snake become an unstoppable python of destruction due to it not being possible to trip it?
If the answer to these questions is "no", simply state (as a DM) "you cannot trip a snake" and move on. If it turns out you ruled incorrectly (due to game balance reasons) you can fix it later.
I already mentioned that "knocking prone" is one of the ways to knock non-hovering creatures out of the air. As a DM you should be aware of this, and allow a flying creature to be "knocked prone" and hence fall - flying in combat in 5e is supposed to be dangerous to make up for its extremely powerful advantages. Creatures with "hover" ignores this, of course.
While a snake is indeed lying on the ground, I wouldn't have the other mechanical effects of prone apply - halved movement speed, advantage and disadvantage on attacks, etc - because the Snakes stats already reflect what it is like to attack a Snake in its standard configuration.
5e, unlike 4e, doesn't have explicit fluff distinct from rules text. The fact that knocking a creature prone actually makes them prone is part of the rules text, it is not fluff you are supposed to ignore.
Similarly the creature is "snake", not "creature with these game statistics with the word 'snake' taped to it". The 5e rules are some of the rules how to model the snake in the 5e combat engine, it doesn't mean you should ignore the fact the we are talking about a snake.
Try to start with "its a snake" when reasoning about the snake, not "it has these game statistics".
(The one big exception is that "an attack that defeats the creature" should mainly focus on depleting HP, and that non-lethal attacks and damage can represent things like "I grab the snake by its head so that it cannot strike".)
You could just houserule that Snakes get Immune: Prone in their stat block. That is fine.
You could also allow snakes to be knocked onto their back and call that "prone". This runs into the twofold problem that a snake on its back is harder to hit with ranged weapons, and that in reality Snakes can flip over much easier than (say) a human can get up from lying on its back.
