I think your view of the verb construction is not correct. "vulgar" is an adjective, so it can't be an object.
The sense of the sentence is: He is vulgar, so I think. This was formulated with an ati-construction (accusative + to-infinitive):
- 1 I think him to be vulgar. - Compare: I hold him to be a good man/I imagine him to be a good man.
And "to be" was dropped:
In 1 vulgar is a predicative complement after the copula verb/linking verb to be. In 2 vulgar is an object complement modifying him.
That's the way I would see the sentence construction.
OALD has a verb pattern/construction of the type: to think sb/sth + adjective - I think it highly unlikely that I'll get the job. - She thought him kind and generous.
See OALD think in no. 1 http://www.oxfordlearnersdictionaries.com/definition/english/think_1?q=think
Of course, you can say it simpler: I think he is vulgar.