Is the room being dusty because of the vacuum cleaner?

Literal

Room [subject-が] dusty [nominalizer-の][topic-は] vacuum-cleaner [possessive-の] fault?

ほこりっぽい is ほこり 'dust' + っぽい, the suffix meaning '-ish, tending to, full of.' The [X]のは[Y]のせい pattern means 'the reason X is because of Y / X is Y's fault.' The implication is the vacuum is malfunctioning. Bare question mark ending — casual, particle-dropped question.