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I studied in Brisk in Yerushalyim and what I saw from my Rosh Yeshiva and his family was that they use regular ashkenazi siddurim but by heart they incorporate their own nuschaos and minhagim (ie. most often following the Gr"a, for example, ommitting נא in the blessing of rachem in bentching. Also ברחמיו in Bonei Yerushalayim is omitted. However, they don't follow every addition/subtraction of the Gr"a. For example, they do not add a חתימת הברכה for Modim Derabanan or add נחם in place of רחם in Shabbos Bentching.

In summation there is no officially sanctioned siddur that is used by Briskers, rather the Mesora they have is incorporated when reciting it.