Which, if any, of the world's languages have both the following features?
- Syllable-initial vowels are disallowed; all syllables must begin with a consonant.
- There is no glottal stop phoneme.
As reported by Lin (1977) (Phonology 14:403-436), there is no glottal stop and no initial vowels in Piro. If you can get Thargari Phonology and Morphology (T. Klokeid 1969, Pac. Ling. Series B #12), you can confirm or deny whether that too is an example.
There's an Australian language Lardil that has both of the features the OP is asking for: