A -->
Correct-
its key successes had been the recent
As there has been a chronological dependency between "well-received beta launch" and "the Internet startup claim".So Past perfect "its key successes had been the recent" is Perfect.Secondly the next dependent sentence starts with a Simple past "the Internet startup claimed" which is grammatically correct.Also usage of Adjective "recent" is correct for the noun clause "well-received beta launch".
B
its key successes has been the recently
Incorrect usage of tense (present perfect) and adverb "recently" doesn't fit in.
C
its key successes is the recently
Incorrect usage of tense (present continuous) and adverb "recently" doesn't fit in.
D
their key successes is the recent
Incorrect pronoun "their" for noun "Internet startup".Also wrong usage of tense (Simple present).
E
their key successes had been the recent
Incorrect pronoun "their" for noun "Internet startup".