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.