Would the term in-cabin be hyphenated or not? As in, the dogs must ride in-cabin on the airplane.
If it is used like an adjective would be, then it should definitely be hyphenated. If used like an adverb would be, then either hyphenated or non-hyphenated is probably fine, though my impression is that the hyphenated use might be more common.
As Edwin Ashworth pointed out, dictionaries do not record in-cabin as a compound word, but there are plenty of attested examples where it is used like one. I will provide some below.
The key question is whether there is any alternative to interpreting in-cabin/in cabin as a compound word. The obvious one (which is really the only one) is that it might be better read as a prepositional phrase (PP). If it is indeed better read as a PP, then it shouldn't be hyphenated. But if there is no possibility to read it as a PP, then it definitely should be. And if it can be read as either, then either hyphenated or not hyphenated is fine.
When used like an adjective would be
Normally, PPs cannot function as pre-head modifiers in noun phrases (NPs), and so we must consider in-cabin as a compound adjective in such cases:
In the above, in-cabin is used attributively, but there are even instances where it is used as a predicative complement:
When used like an adverb would be
PPs routinely function as adverbials: put it i͟n͟ ͟t͟h͟e͟ ͟c͟a͟r͟, traveled b͟y͟ ͟t͟r͟a͟i͟n͟, located o͟n͟ ͟y͟o͟u͟r͟ ͟l͟e͟f͟t͟. The only issue with in cabin is that cabin is a count noun in the singular, so we would normally expect a determiner in front of it, like this:
Pets up to approximately 5 kg are allowed to travel i͟n͟ ͟t͟h͟e͟ ͟c͟a͟b͟i͟n͟. (source)
However, there is precedent for dropping the determiner in PPs. For example, we say that we travel b͟y͟ ͟b͟u͟s͟, go t͟o͟ ͟c͟h͟u͟r͟c͟h͟, appear o͟n͟ ͟t͟e͟l͟e͟v͟i͟s͟i͟o͟n͟, talk b͟y͟ ͟t͟e͟l͟e͟p͟h͟o͟n͟e͟, etc. What is common to all these cases is that the noun doesn't refer to a particular object but to some generalized notion, a type: a type of transport, institution, media, and communication, respectively.
So perhaps we can say travel in cabin and mean a type of travel: one where e.g. a dog rides in the cabin as opposed to in the storage area.
There are attested examples of such usage:
He wished to ride i͟n͟ ͟c͟a͟b͟i͟n͟. (source)
On the other hand, there is also precedent for combinations preposition+noun becoming compound adverbs:
(Note however that there is also a PP version of the latter: occurring, as it does, o͟f͟f͟ ͟s͟t͟a͟g͟e͟ (source). But the OED does record offstage as an adverb, where some of the examples of such usage that it provides use the hyphenated form off-stage. Confusingly, however, some of its examples leave it as two non-hyphenated words. With all due deference to the authority of the OED, I would suggest these cases have been mischaracterized and that off stage is a PP rather than a single compund adverb there.)
And indeed, many sources do use in-cabin as a compound adverb:
Korean Air allows small pets to travel i͟n͟-͟c͟a͟b͟i͟n͟ for $200 a piece. (source)
Dogs who can comfortably fit under the seat are permitted to ride i͟n͟-͟c͟a͟b͟i͟n͟. (source)
Your furry friend is too large to travel i͟n͟-͟c͟a͟b͟i͟n͟. (source)
For animals that are not traveling i͟n͟-͟c͟a͟b͟i͟n͟, the ARK strongly recommends... (source)
Therefore, it seems that when in-cabin/in cabin is used like an adverb would be, both the hyphenated and non-hyphenated versions are OK.