eqnofmotion := 1/r^3 D[r^3 D[f[r], r], r] + M0^2/(r^2 + L^2)^2 f[r]
L := 1
M0 := 2 Sqrt[2]
sltn = NDSolve[{eqnofmotion == 0, f[10^-5] == 1, f'[10^-5] == 0},
   f, {r, 10^-5, 10^2}];
Plot[ f[r] /. sltn, {r, 10^-5, 10^2}, PlotRange -> {-1, 1},
 PlotStyle -> {Thick, Black},
 BaseStyle -> {18, FontFamily -> "Times New Roman"},
 AxesLabel -> {"[Rho]", "[Delta]L([Rho])"},
 PlotLegends -> Automatic]
f[r] /. sltn /. r :> 10^2

function0[r_] := f[r] /. sltn
derivative0[r_] := function0'[r]
function0[r] /. r :> 100
derivative0[r] /. r :> 100

scalarasymptote[r_] := operator/r^3 + source r
(*Define and solve the system to determine operator and source*)

Solve[{scalarasymptote[100] == 0.0000999607,
  scalarasymptote'[100] == - 1.99889 10^-6}, {operator, source}]

(*Now use that [ScriptCapitalJ] = g^2/[CapitalLambda]^2
[ScriptCapitalO]*)
(*[CapitalLambda] is the UV cutoff.*)
[CapitalLambda] := 10^2
NSolve[ 2.49983 10^-7 == 74.9624 a/[CapitalLambda]^2, a]

eqnofmotion[L_, M0_] :=
  1/r^3 D[r^3 D[f[r], r], r] + M0^2/(r^2 + L^2)^2 f[r];
scalarasymptote[r_] := operator/r^3 + source r;

sltn[L_, M0_] :=
  NDSolve[{eqnofmotion[L, M0] == 0, f[10^-5] == 1, f'[10^-5] == 0},
   f, {r, 10^-5, 10^2}];

sltn2[L_, M0_] :=
  Solve[Evaluate[{scalarasymptote[100] == f[100],
      scalarasymptote'[100] == f'[100]} /.
     First@sltn[L, M0]], {operator, source}];

sltn3[L_, M0_, Λ_] :=
  Solve[(source == operator a/Λ^2) /.
    First@sltn2[L, M0], a];

Plot[a /. sltn3[1, m, 100], {m, 0, 2 Sqrt[2]}]