An alternative approach is to use ParametricNDSolveValue with the `second argument set to the function you want to plot:
ClearAll[pndsv]
pndsv = ParametricNDSolveValue[{r''[t] == r[t]*ϕ'[t]^2 - 1/r[t], ϕ'[t] == d/r[t]^2,
ϕ[0] == a, r[0] == b, r'[0] == c},
{r[#] Cos[ϕ[#]], r[#] Sin[ϕ[#]]} &,
{t, 0, 200},
{a, b, c, d}];
params = Transpose[{vTangential/r0, r0, vRadial, L}];
ParametricPlot[Evaluate[pndsv[##][t] & @@@ params], {t, 0, 2 Pi},
GridLines -> Automatic, Frame -> True]
Interactively set up to 10 sets of parameters:
k = 10;
Manipulate[ParametricPlot[Evaluate[pndsv[##][t] & @@@ Take[psets, n, 4]], {t, 0, 2 Pi},
GridLines -> Automatic, Frame -> True, ImageSize -> 400, AspectRatio -> 1],
{{psets, ConstantArray[1., {k, 4}]}, None},
{{n, 3}, 1, 10, 1},
Dynamic[Panel[Grid[Prepend[#, {"params", "a", "b", "c", "d"}] &@
MapIndexed[Prepend[#, #2[[1]]] &, Outer[Manipulator[Dynamic[psets[[#1, #2]]], {0, 3},
Appearance -> "Labeled", ImageSize -> Tiny] &, Range[n], Range[4]]],
FrameStyle -> LightGray,
Background -> {None, None, {# + 1, 1} -> ColorData[97]@# & /@ Range[n]},
Dividers -> {{False, True}, {False, True}}]]],
Alignment -> Center]
