Can a prepositional phrase starting with "during" work as an adjectival phrase?

As noted in comments, a prepositional phrase using during can be used to modify a noun. The example used during the day to modify work used as the subject.

The problem with your sentence lies elsewhere: it’s that during doesn’t work very well for plans. One has plans for one’s vacation, not “during” it.

Plus during the vacation sounds funny for other reasons. It is hard to construct a context where the vacation works at all; it doesn’t work like the holidays might. I suppose you could say during the summer vacation to mean over the summer vacation, but I would leave out that the myself and just say over summer vacation.

For example, these all work:

• I have plans for doing something special during my vacation this summer.

• I have plans to do something special during my vacation this summer.

• I plan to do something special during my vacation this summer.

• I plan to do something special over summer vacation.

"A prepositional phrase comprising a preposition and a noun phrase can generally function either as an adjectival phrase or as an adverbial phrase."

My use of the term, a complement only describes the relation one phrase has to another. It has nothing to do with whether that phrase is required syntactically.

Generally, prepositional phrases function as noun complements or verb complements. In either case their relation to the noun or verb is descriptive. Your terms, adverbial phrase and adjectival phrase make the same descriptive distinction but confuse semantic relations (e.g. adverbial/ adjectival = descriptive relation) with syntactic ones (e.g. noun complement or verb complement), of which both have descriptive relations to their antecedents.

The term adverbial (at least in the way I use it and understand it) indicates a type of relation that a syntactical element takes to a verb. Adverbials generally describe where, when why or how an action or state (verbs) happens. Syntactically, adverbials are just verb complements. Objects are verb complements too, but they have an object relation to the verb, saying what thing is effected by the action or state. Subject complements are also verb complements but only syntactically; semantically they have a descriptive relation to the subject. Adverbials and object are linked semantically and syntactically to the verb. All verb complements in English can be described with one of these three relations but they are still just verb complements.

So,

The book on the table is mine. S[NP+nc(pp):book] | V | SC(np:pn)

I put the book on the table. S|V|O(np)+A(pp)

Do you have any plans for your vacation? S|V|O(NP)+a(pp) // S|V|O[NP+nc(pp):plans]

The last sentence seems to have two possible parses, which equate to the same interpretation. In fact the pp is doing both of these things - it is a verb complement, while at the same time it is a noun complement. This is pretty common and a source of some argument. Most paring systems only allow for a phrase to function in one way, even when the phrase is playing more than one role in a sentence.