Patrick,
You need parenthesis around expressions when you use &&. It takes priority over other operators.
RegionPlot3D[(-96.3 <= x <= 96.3) && (Sqrt[3]/3*Abs[x] - 103.74 <=
y <= 103.74 - Sqrt[3]/3*Abs[x]) && (0 <= z <=
Sqrt[(103.74 - 1/3 Sqrt[3] Abs[x])^2 - y^2]), {x, -97,
97}, {y, -104, 104}, {z, 0, 104}]
Regards