Are verbal nouns or suru verb transitive, intransitive, both, or either depending on the verbal noun or suru verb? For example: if you had the suru verb 勉強する (study), could it have a direct object and therefore be intransitive/transitive?
If verbal nouns can be either transitive or intransitive, where can I find if a verbal noun or suru verb is transitive/intransitive? Jisho dictionary doesn't show if suru verbs are intransitive/transitive.
I sense some confusion around the terms transitive and intransitive. Let's define our terms first, and then we can look at your question.
In English grammar contexts, transitivity is often talked about as a property of syntax -- the structure of the given sentence. A verb is "transitive" if that sentence has an explicitly stated object for that verb. Meanwhile, a verb is "intransitive" if that sentence does not have any explicitly stated object for that verb.
That's actually what the label transitive refers to: the action of the verb transits or "goes across" to some other noun.
Compare:
In Japanese grammar contexts, transitivity is a instead property of semantics -- the meaning of a given verb. A verb is "transitive" if that verb describes an action on some other noun. Meanwhile, a verb is "intransitive" if that verb does not describe an action affecting some other noun.
(Note that what specifically meets the terms of "affecting some other noun" can be culturally bound and tricky to figure out -- more on that below.)
The Japanese terms for these are 他動詞【たどうし】, literally "other-acting word", and 自動詞【じどうし】, literally "self-acting word".
Compare:
Are verbal nouns or suru verbs transitive, intransitive, both, or either depending on the verbal noun or suru verb?
Herein lies some confusion.
I'll ignore the bit about "verbal nouns", since a noun by definition cannot be transitive or intransitive.
Restated, your question amounts to, "is a verb transitive depending on the verb?" In which case, the answer has to be yes.
However, I think instead you're asking, "How do I tell if a suru verb is transitive?" There are two ways to approach this.
Japanese dictionaries do a sadly awful job in recording this information. Some dictionaries will at least indicate whether a given suru verb is even a suru verb, and not just a noun -- Daijirin does this, as does Digital Daijisen: here's Daijisen's entry for 学習【がくしゅう】, for instance. The top of the entry has the notation ［名］(スル), indicating that this is a [名]{めい}[詞]{し} or noun, and that it can take する as a verb. But there's no explicit indication whether this is a 自動詞【じどうし】 (intransitive verb) or 他動詞【たどうし】 (transitive verb). We have to look at the sample questions to guess that it might be a 他動詞【たどうし】 (transitive verb), since one of these uses the direct-object construction 「英語[を]{●}学習する」.
By contrast, here's the 学習【がくしゅう】 entry in Shogakukan's Kokugo Dai Jiten. That entry only indicates 〘名〙, that the word is a [名]{めい}[詞]{し} or noun, with no information about verb-ness or transitivity -- not even any sample sentences using a verb construction.
Another way of approaching this question is to consider the semantics of the verb, the underlying meaning: Does this action require an object in order to make sense?
Think about "eating" and "walking" in our examples above. Then think about suru verbs like 勉強【べんきょう】する ("to study") or 学習【がくしゅう】する ("to learn"). Is it possible for someone or something to 勉強【べんきょう】 or 学習【がくしゅう】 without an object? No. These actions fundamentally require an object, even if that object is unstated. We know that when someone 勉強【べんきょう】するs or 学習【がくしゅう】するs, they are doing those actions on something else -- whatever books or subject material they are studying or learning from. So we can surmise that these suru verbs are 他動詞【たどうし】 or t̲r̲a̲n̲s̲i̲t̲i̲v̲e̲ verbs.
What about 故障【こしょう】する ("to have a breakdown, to malfunction")? Is it possible for someone or something to 故障【こしょう】する without an object? Yes. This action is fundamentally something that a noun does all on its own. We know that when something 故障【こしょう】するs, that something is doing the action itself -- nothing else is directly affected. So we can surmise that this suru verb is a 自動詞【じどうし】 or i̲n̲t̲r̲a̲n̲s̲i̲t̲i̲v̲e̲ verb.
As we see above with our 歩【ある】く example, some 自動詞【じどうし】 or i̲n̲t̲r̲a̲n̲s̲i̲t̲i̲v̲e̲ verbs can still take direct objects. In many such cases, these are verbs of motion, and the syntactical objects describe where or how the action is being performed. Consider English constructions like "drive the highway", or "go that route", where highway and route are the objects of the verbs.
And some verbs are a̲m̲b̲i̲t̲r̲a̲n̲s̲i̲t̲i̲v̲e̲, where they describe action that can be either "self-acting" or "other-acting". Take 完成【かんせい】する ("to complete"), for instance. Daijisen's entry includes sample sentences showing both 自動詞【じどうし】 and 他動詞【たどうし】 senses:
Meanwhile, this Chiebukuro thread suggests that at least some native speakers consider 完成【かんせい】する to be exclusively a 自動詞【じどうし】 / i̲n̲t̲r̲a̲n̲s̲i̲t̲i̲v̲e̲ verb. So usage patterns for this particular verb may not be entirely settled.
"How do I tell if a suru verb is transitive?"
Ultimately, 1) look in dictionaries for sample sentences, 2) think about the nature of the action itself, and 3) also check in with native speakers.
Another quicker answer.
As you might have noticed, 勉強する can be both transitive and intransitive.
There are many such suru-verbs (with transtitive/intransitive uses), this article contains a list for example. Note that there are some disagreements among dictionaries.
学研国語大辞典 seems to contain explicit information on transitivity. (I have no idea on the legitimacy of the website.)
