Just Set dist[[1]] to your output inside Manipulate. Putting additionally a Dynamic to your plots let's you see the change live. So after evaluating all your definitions from the first code-block, you could do something like
Column[{
Manipulate[
distToMean[[1]] = (Flatten@
allTeamAverageStats[[Flatten@
Position[
Total[{v1, v2, v3, v4, v5, v6, v7, v8, v9, v10}/
Total@{v1, v2, v3, v4, v5, v6, v7, v8, v9,
v10} (allTeamAverageStats[[#]] - rosterRAMean)/
rosterRASD] & /@ Range[792],
Max[Total[{v1, v2, v3, v4, v5, v6, v7, v8, v9, v10}/
Total@{v1, v2, v3, v4, v5, v6, v7, v8, v9,
v10} (allTeamAverageStats[[#]] - rosterRAMean)/
rosterRASD] & /@ Range[792]]]]]) - rosterRAMean, {v1,
Range[.1, 1, .1]}, {v2, Range[.1, 1, .1]}, {v3,
Range[.1, 1, .1]}, {v4, Range[.1, 1, .1]}, {v5,
Range[.1, 1, .1]}, {v6, Range[.1, 1, .1]}, {v7,
Range[.1, 1, .1]}, {v8, Range[.1, 1, .1]}, {v9,
Range[.1, 1, .1]}, {v10, Range[.1, 1, .1]},
ControlType -> Manipulator],
Dynamic[
Show[Function[attributes,
Graphics[{Blend[{{-Max[Abs[distToMean[[1]]]], Red}, {0,
LightRed}, {0, LightGreen}, {+Max[Abs[distToMean[[1]]]],
Green}}, distToMean[[1]][[attributes]]],
Rectangle[{If[distToMean[[1]][[attributes]] < 0,
distToMean[[1]][[attributes]]*10, 0],
attributes - 1}, {If[distToMean[[1]][[attributes]] < 0, 0,
distToMean[[1]][[attributes]]*10], attributes}],
PlotRange -> {{-6, 6}, {0, 10}}}, AspectRatio -> 16/9,
Epilog -> {Black, Thickness[.05], Line[{{0, 0}, {0, 11}}]}]] /@
Range[10], Frame -> False, FrameTicks -> False]]
}]
Note: This is far from being good programming style but for a test it should be fine.