That depends on how you're asking it!
If you mean 'vague' as in the word itself, it is an adjective. According to Dictionary.com, their definition is "not clearly or explicitly stated or expressed."
If you mean 'vague' as in - what part of speech is broad or not thoroughly detailed, that would be articles. A commonly used article would be 'The'. The reason that this is the most vague part of speech is that there are very few articles and not many rules follow articles in comparison to the other parts of speech.
If you do not consider 'articles' one of the parts of speech in English, then the most vague part of speech would be interjections. Interjections are used to express emotions. Words such as 'Wow' or 'Yippee' are interjections. Also, filled pauses such as uh, er, or um are also interjections. Interjections are usually followed by exclamation marks. (Remember, almost any word can be made an interjection given enough emphasis.)
adverb
what part of speech is beneath
i want to know what part of speech is camping
The part of speech is a adjective
The part(s) of speech that has an "A" sound are: Adjective and Adverb.
a verb
a verb
Wicked is an adjective.
The letter D represents a determiner on the diagram. Determiners are words that introduce and clarify nouns.
part of speech
The part of speech for this particular word is a noun.
The part of speech for "answer" is a noun.
adverb
what part of speech is beneath
The part of speech for "explicit" is an adjective.
The part of speech for "twirl" is a verb.
The part of speech for "unfamiliar" is an adjective.