I'm wondering why we use で with 休暇 and why it can't be に like in 夏休みに.
Update: In the sentence,「私たちは休暇でここに来ました。海辺のホテルに泊まっていました。」, I wondered why they used で instead of に.
In fact, both can be correct Japanese, if the context allows. But I guess, in most cases, what you want to say is the former one.
"We were here for vacation. We were staying at a hotel by the sea."
It will be ungrammatical if you interpret this で as a time/space marker, because 休暇 is neither a place nor a one-time event. Instead, it should be understood as reason or means. Therefore 休暇で means "because of vacation" or "making use of vacation", which you'd put in English for/on vacation.
"We were here during a vacation. We were staying at a hotel by the sea."
One possible interpretation of に here is "at", in other words, a certain point in time or space. Then the sentence consequently can be only a remark about a past thing (Japanese -た itself could be either past or perfect), because it is not possible to conceptualize an event you are still involved in as a timeless point.
"We were here in order to have a vacation. We were staying at a hotel by the sea."
There is another possibility, that the に means objective. This will just sound as much as the English sentences, so... may be usable under some situations, for example, when they're asked about the purpose of their trip, or when they have little choice but to come here if they want to take a vacation etc.
I see no evidence for what you suggest. Do you have the full sentence that you are referencing?
Edit: As I was getting downvotes, I've decided to copy/paste my answer here.
休暇で means 'for break/for vacation', rather than 'during (on) break/during (on) vacation'. 私たちは休暇に入りました would be 'We went on vacation.'. This would be the same structure as in the example link I posted 休暇に入る, to enter a vacation period. As I haven't studied the language in a while (just used it daily), I can't explain the difference well grammatically. Go to ejje.weblio.jp/sentence and type in 休暇で and 休暇に and it will give you many example sentences. Learning by real usage is usually the best way to get it to click.
That being said, 休暇中に might be easier to use for 'during the break'.