Is it possible to shoot a goodberry from a blowgun using the rules as written?
The reason for my hesitation is because a blowgun hole might be too small for the goodberry to fit in, but then again, I don't know the size of a goodberry.
If it indeed can be shot from a blowgun, could it be shot in the mouth of an unconscious ally in order to heal him?
The Blowgun lists Ammunition as a property which gives several choices in the PHB. While we understand the PHB is not an exhaustive list, we generally accept that variations on it are thematic and functional and not mechanical in nature.
Would definitely not be a Goodberry. The Goodberry does not have the same type of aerodynamic properties that a dart has, nor is it a type of ammunition in general.
5e also doesn't have called shots as a standard option or variant rule that I have found. Being able to shoot into their mouth is not an available RAW option.
Allowing this would likely fall under a houserule in order for both the goodberry mechanic and called shot mechanic. You still can 'feed' someone in a goodberry who is unconscious much like you can have them drink a potion. Unlike the wording in Healing Potions, Goodberry only states that it takes an action to consume (with no mention of administering.) However, Jeremy Crawford believes you can administer it to an unconscious character.
As another strike against ranged Goodberries, spending an action to feed an unconscious character is very different than spending an action to remotely delivery a substance into the mouth of an unconscious character and have it be consumed.
But turning this into a Healing Word distance heal greatly increases the power of goodberry or administering to fallen allies in a system where healing is generally a touching experience and distance healing is only an option in a couple of circumstances.
If the DM rules that the berry is too large then this can't even be
attempted, or is an auto fail1 with the blowgun.
You could try an Athletics check (Ability check, Strength) something
similar to trying to hit a long three point shot, or to hole out a
chip shot(golf), with the abilty check attempt being phrased
something like "toss the goodberry into the ally's open mouth" ...
see the footnote for details.
If it is ruled to fit ...
Blowguns have an ammunition property, which is a particular thing in the game, and in which category a goodberry is not listed. Since this item isn't blowgun ammunition, proficiency with a blowgun should not/will not not apply (one may ask the DM for a favorable ruling on that, but not expect it).
Ability Checks
An ability check tests a character’s or monster’s innate talent and training in an effort to overcome a challenge. The DM calls for an ability check when a character or monster attempts an action (other than an attack) that has a chance of failure. When the outcome is uncertain, the dice determine the results. For every ability check, the DM decides which of the six abilities is relevant to the task at hand and the difficulty of the task, represented by a Difficulty Class. Basic Rules p. 58)
Depending upon the circumstances of this attempt, you can expect that the DC will be set somewhere from Hard to Very Hard to Nearly Impossible: a DC between 20 and 30.
\begin{array}{l|l} \text{Task Difficulty} & \text{DC} \\ \hline \text {Very Easy} & 5 \\ \text {Easy} & 10 \\ \text {Medium} & 15 \\ \text {Hard} & 20 \\ \text {Very Hard} & 25 \\ \text {Nearly Impossible} & 30 \end{array}
Circumstantial disadvantage may apply: that depends on if it's in the middle of combat, or in a non-combat situation where you can't reach the other character. (for whatever reason)
Advantage and Disadvantage
The DM can also decide that circumstances influence a roll in one direction or the other and grant advantage or impose disadvantage as a result. (Basic Rules p. 57)
The two most likely choices are Athletics (effective use of breath/power) or a Dexterity check (for aim). Non-proficiency is the rational ruling here, but perhaps your cleric can cast the guidance cantrip to increase the odds on the attempt by a 1d4, or the DM can rule within RAW that (if you are proficient in Athletics) that proficiency would apply. If the attempting character is a 7th level or higher Champion(Fighter), the class ability Remarkable Athlete offers a chance for a point or two (1/2 proficiency) bonus for ability checks that you are not proficient in if they are Dexterity, Constitution, or Strength based ability checks. (Basic Rules, p. 26)
By RAW, ability checks are a thing, and by RAW this is how the game is played:
1Going back to the opening sentence of this answer: if the first time you try this the DM rules that "goodberries don't fit in a blowgun" then it's an auto-fail, no roll required, since there is no uncertainty regarding the outcome.
Don't try to use the blowgun, try to toss the goodberry into the open mouth. The process is the same: the DM sets a DC (which will likely be pretty high) and you make an ability check (probably athletics) rather than an attack.
As @Miniman points out, there isn't an explicit mechanic for "feeding a goodberry to another character" but there isn't an explicit prohibition against it either. As @V2Blast points out, Jeremy Crawford has said he'd allow it (as DM) ... thus the rules lack specificity on the ultimate outcome of such an attempt. The question then becomes ... how does that character expend an action to consume the goodberry, if that PC is at 0 HP/unconscious?
Regardless of how one tries to get the goodberry into the mouth of the unconscious character, the character still has to spend an action to consume the goodberry.
A creature can use its action to eat one berry. Eating a berry restores 1 hit point ... (PHB, p. 246)
Unconscious creatures can't perform actions due to being incapacitated.
An incapacitated creature can't take actions or reactions.
An unconscious creature is incapacitated ...
(PHB, Appendix A; p. 290 and 292; Conditions; Unconscious; Incapacitated)
Basically, you can try to do this, but it might not have the effect that you want it to. If you can figure out a way to get the unconscious character to spend an action to eat the goodberry, the other means of trying to get the goodberry into the character's mouth may provide an opportunity to get your plan to work: heal that 1 HP and get the character up and moving.
RAW provides neither the physical dimensions for a Goodberry, nor the physical dimensions of a blowgun. Thus, there's no way to rule RAW whether this would work or not.
As DM, the way that I tend to imagine Goodberries is that they look like cranberries, so I'd probably allow this to work, with the caveat that I'd require an attack roll with the blowgun contested by an AC of 15 to successfully land the shot in the target's mouth. Roll would otherwise be treated as a normal attack roll, so the shooter might gain advantage or disadvantage depending on the circumstances.
In your case specifically, the disadvantage from the target being prone is cancelled by the advantage of the target being unconscious, and thus it would be a normal roll.
Alternatively, if calling it an "attack roll" implies too many combat-oriented mechanics for this kind of roll, then I'd consider just making it a Dexterity roll with a DC of 15, adding proficiency if the user has proficiency with blowguns.
