I saw the last response after recoding to get something workable. I thought I would show it so that there is a complete solution in the responses.

khat = .162;
xin = .028;
vhat = .017;

WronsFUN[\[Lambda]_?NumberQ, w_?NumberQ, xoo_?NumberQ] := Module[
  {val, nIN, nOUT},
  nIN = NDSolve[{\[Lambda] == -I*w*x^(3/2) - vhat*x/y[x]*y'[x] - 
       7/2*khat*x^2/y[x]*y'[x] - khat*x^3/y[x]*y''[x], 
     vhat*xin^(-1/2)*y[xin] + khat*xin^(3/2)*y'[xin] == y[xin], 
     y[xin] == 1}, y[x], {x, xin, 1}];
  yIN[x_] = First[y[x] /. nIN];
  nOUT = NDSolve[{\[Lambda] == -I*w*x^(3/2) - vhat*x/y[x]*y'[x] - 
       7/2*khat*x^2/y[x]*y'[x] - khat*x^3/y[x]*y''[x], 
     vhat*y[1] + khat*y'[1] == -y[1], y[1] == 1}, y[x], {x, xin, 1}];
  yOUT[x_] = First[y[x] /. nOUT];
  val = yIN[xoo]*yOUT'[xoo] - yOUT[xoo]*yIN'[xoo]; val]

With those definitions the following now seems to work as expected. Or at least not complain about working conditions, salary, or the like.

FindRoot[WronsFUN[j, 21.67, .038], {j, .1 - .1*I}]

The problem happens before you ever get to FindRoot

In[1]:= khat = .162; xin = .028; vhat = .017; {\[Lambda], w, xoo} = {j, 21.67, .038};
  NDSolve[{\[Lambda] == -I*w*x^(3/2) - vhat*x/y[x]*y'[x] - 7/2*khat*x^2/y[x]*y'[x] - khat*x^3/y[x]*y''[x], 
     vhat*xin^(-1/2)*y[xin] + khat*xin^(3/2)*y'[xin] == y[xin], y[xin] == 1}, y[x], {x, xin, 1}]

During evaluation of In[1]:= NDSolve::ndnum: Encountered non-numerical value for a derivative at x == 0.028`. >>

Out[4]= NDSolve[{j == (0. - 21.67 I) x^(3/2) - (0.017 x Derivative[1][y][x])/y[x] - (0.567 x^2 Derivative[1][y][x])/y[x] - (
    0.162 x^3 (y^\[Prime]\[Prime])[x])/y[x], 0.101594 y[0.028] + 0.000759018 Derivative[1][y][0.028] == y[0.028],
   y[0.028] == 1}, y[x], {x, 0.028, 1}]

So NDSolve is failing with the very first point. Study that until you can resolve the problem.

You really should reformat so that input is clearly demarcated and easy to cut and paste. Also you might do better by making the function a "black box" that only evalautes on explicit numeric input. This can be done like so.

WronsFUN[\[Lambda]_?NumberQ, w_?NumberQ, xoo_?NumberQ] := ...

Don't forget to first Clear that function so as to remove the existing definition.