A Yahoo answer says no, but I thought I'd ask my more enlightened StackExchange friends.
Bulgaria is very close to the native word: България (pronounced /bəlˈɡarija/). This name has a Turkic origin, which fits with the history of the Bulgar people, who were a group of Turkic tribes who settled in a region where the dominant language family was Slavonic and adopted that language. Wiktionary in Bulgarian (yes, I know) lists Turkic cognates, and Wiktionary in English lists a Turkic origin. Wikipedia notes that the etymology is disputed, but does not offer a non-Turkic hypothesis. A reference found by Kenny LJ mentions that a cognate of Bulgar was used in China in the 1st century BC, well before the Romans may have had a chance to name them.
The root of vulgar is attested in Latin well before the Bulgars migrated into the Balkans (as in vulgus, the people, the masses).
Given that Turkic and Indo-European languages are not known to be related, the likelihood that there is a common origin to the two words is extremely small, and if there was one it would be extremely remote.
It is very likely that the case is coincidence Bulgar-Vulgar. But giving the fact that the Byzantines \At that time writing in roman, the main source for the Bulgarians from the very first appearance of them on the Balkans in 5 century BC\ always describe them in negative way, or just with neglect for the real identity of the tribes, it is very plausible the use by them of the word 'Vulgus' and 'vulgarian' just by the way. Also it`s interesting that there is no mention of tribe 'Bulgar/Bolgar' during the settlement south of Danube in 680 BC. The Byzantines sources explicitly point 'Kutrigurs, Onogurs, Utigurs" and others but Bulgars. It is true that they very positively described their country 'Old Great Bulgaria" with the proven christian ruler - Khan Kubrat as their and personal friend of the Emperor Irakliy with whom they have had life-long peace and alliance, after which the settlement had begun. After that when the hostilities had been the agenda and until today some Byzantines and Greeks intentionally had errored the first letter 'B" with 'V", common and distinctive feature between neighbours.