Strictly speaking, the Torah mandates thirty Shofar blasts on Rosh Hashana. In order to ensure that we have unquestionably fulfilled this requirement, the current practice is to hear one hundred blasts; thirty when sitting and listening to the Shofar, thirty during the silent Musaf Amida, thirty during the repetition of Musaf and ten at the end of the prayer. The Shulhan Aruch (Orah Haim 592:1) rules that the Shofar should be blasted during the repetition of Musaf and makes no mention of the silent Amida, because they may distract the congregants while praying silently. The original Sephardic custom was to only blow the Shofar during the repetition of Musaf, and this is still the custom of Meknes as well as some Ashkenazic communities.
One explanation of why the blasts were incorporated into the silent Amida is that, since the Moroccan custom is for an assistant Hazan to recite the Amida aloud while everyone is praying silently anyway, there is no concern that the Shofar will be distracting. Additionally, the Kabbalah attributes great importance to the blowing the Shofar during the silent Amida. As such, this is the current Sephardic and Hasidic custom and that of certain Ashkenazic communities.
Summary: The Moroccan custom is blow the Shofar during both the silent Musaf Amida and its repetition.