I was looking for any official way - class feature, magic item, anything - for a character to add their proficiency to any form of Constitution check, but I couldn't find anything. Is it possible to do this?
Ability Check Proficiency is a variant rule.
There is a variant rule in the Dungeon Master's Guide (Page 263) that replaces skill proficiency entirely:
Ability Check Proficiency
With this variant rule, characters don’t have skill proficiencies. Instead, each character has proficiency in two abilities: one tied to the character’s class and one tied to the character’s background. The Ability Proficiencies by Class table suggests a proficiency for each class, and you choose which ability is tied to a given background. Starting at 1st level, a character adds his or her proficiency bonus to any ability check tied to one or the other of these two abilities.
If you aren't using this variant option, then proficiency is explicitly tied to skills, rather than abilities.
There are a couple of half of an exceptions to this. The bard's 2nd level Jack of All Trades feature allows the bard to add half their proficiency bonus to any ability check. The Champion Fighter has a 7th level feature, Remarkable Athlete, that allows them to add half their proficiency (rounded up!) to Constitution checks. These cannot be used at the same time.
Your DM needs to call for a Constitution Ability Check that uses one of your skill proficiencies
There are sections, both in the Player's Handbook and the Dungeon Master's Guide, that discuss alternative combinations:
Normally, your proficiency in a skill applies only to a specific kind of ability check. Proficiency in Athletics, for example, usually applies to Strength checks. In some situations, though, your proficiency might reasonably apply to a different kind of check. In such cases, the DM might ask for a check using an unusual combination of ability and skill, or you might ask your DM if you can apply a proficiency to a different check. For example, if you have to swim from an offshore island to the mainland, your DM might call for a Constitution check to see if you have the stamina to make it that far. In this case, your DM might allow you to apply your proficiency in Athletics and ask for a Constitution (Athletics) check. So if you're proficient in Athletics, you apply your proficiency bonus to the Constitution check just as you would normally do for a Strength (Athletics) check.
Similarly, when your dwarf fighter uses a display of raw strength to intimidate an enemy, your DM might ask for a Strength (Intimidation) check, even though Intimidation is normally associated with Charisma.
―Variant: Skills with Different Abilities, Player's Handbook, pg 175
As described in the Player's Handbook, a skill proficiency represents a character's focus on one aspect of an ability. Among all the things a character's Dexterity score describes, the character might be particularly skilled at sneaking around, reflected in proficiency in the Stealth skill. When that skill is used for an ability check, it is usually used with Dexterity.
Under certain circumstances, you can decide a character's proficiency in a skill can be applied to a different ability check. For example, you might decide that a character forced to swim from an island to the mainland must succeed on a Constitution check (as opposed to a Strength check) because of the distance involved. The character is proficient in the Athletics skill, which covers swimming, so you allow the character's proficiency bonus to apply to this ability check. In effect, you're asking for a Constitution (Athletics) check, instead of a Strength (Athletics) check.
Often, players ask whether they can apply a skill proficiency to an ability check. If a player can provide a good justification for why a character's training and aptitude in a skill should apply to the check, go ahead and allow it, rewarding the player's creative thinking.
Having proficiency with a tool allows you to apply your proficiency bonus to an ability check you make using that tool. For example, a character proficient with carpenter's tools can apply his or her proficiency bonus to a Dexterity check to craft a wooden flute, an Intelligence check to craft a wooden secret door, or a Strength check to build a working trebuchet. However, the proficiency bonus wouldn't apply to an ability check made to identify unsafe wooden construction or to discern the origin of a crafted item, since neither check requires tool use.
―Proficiency, Dungeon Master's Guide, pg 239
So if you want to apply proficiency or expertise to a Constitution Ability Check, your DM needs to provide a context where it's appropriate to ask for a Constitution Ability Check, and then they will either need to declare that a specific skill proficiency applies, or you (as you are allowed to do by the rules) can ask to apply one of your skill proficiencies to the roll, as appropriate for the check being made and what your character is doing.
There is, however, a big problem, which you might have already realized:
Most DMs think of Ability Checks as "Skill Checks"
From my observations, having played for a bunch of different DMs in 5e, and having listened to a lot of actual play 5e podcasts, one thing I've noticed is that most DMs in 5th Edition D&D still think of Ability Checks as being "Skill Checks", which is a term that has been inherited from older editions of D&D, but doesn't actually exist in 5e.
For example, if a player takes the Hide action, and needs to make a Dexterity (Stealth) check, the DM will say out loud "make a Stealth check", and the player will just use whatever is listed as the modifier for Stealth in the section of their character sheet that lists their skills―which, in this case, automatically assumes that Dexterity is the ability associated to Stealth and that the modifier printed there is what the DM is always going to call for.
And this usually doesn't break down: the game rules themselves, and the officially printed campaign modules both, generally don't call for unusual Ability/Skill combinations. And, both also don't generally call for Constitution Ability checks at all, or associate any skill proficiencies to those checks. As shown above, both the PHB and DMG do offer one example, but it's under a variant rule where it's explicitly noted to be at the DM's discretion, so it's not surprising that it doesn't see much use in play: DMs just aren't trained to think of Ability Checks in this way, and because of the preponderance of DMs who started with earlier editions and still think of them as "Skill Checks", they've taught a lot of newer DMs who might not see those variant rules and consider adding them to play.
So what I'm going to offer below are a couple of examples of Constitution Ability checks I've called for in my campaigns, in the hopes that you or your DM might see them and use them as inspiration to think outside-the-box in ways that the officially printed rules are lacking:
- As per the examples listed in the PHB/DMG, I have used Constitution (Athletics) for characters who need to tread water for hours at a time. One time the party hid in a pool of water while waiting for nearby NPCs to change positions, and the check represented the group not tiring themselves out while in the water.
- I've also used Constitution (Athletics) to represent characters pushing the Travel Pace distance on long travels, where failure might mean slower-than-expected progress or injury resulting from pushing themselves so far.
- I've used Constitution (Acrobatics) to represent a check where a character had to pose in place for a very long time; I probably would have allowed their Performance proficiency instead if they had asked.
- One time, when the party knew that a platoon of soldiers was going to travel down the road, I called for Constitution (Stealth) for the entire party to stay hidden on the side of the road while the soldiers passed. Dexterity wouldn't have been appropriate in this context, since the party had ample time to prepare for their passage, and the difficulty wasn't finding and quickly assimilating into a hidden position, but was being able to hold still in their hidden position for the entire time the soldiers were present and potentially able to be aware of them
- I've also used Constitution (Athletics) to handle characters climbing up multiple dozens of flights of stairs at once. Because Mages have a funny sense of humor about how they build their towers.
Hopefully this is a good vertical slice of the kinds of checks that would make the use of Constitution, and associated skill proficiencies, make sense.