I'd like to extend my comments to an answer. First of all, for those in _v12_ or higher, `FiniteElement` is the best choice for this problem, as shown in [user21's answer](https://mathematica.stackexchange.com/a/219608/1871). But, if you're in a version lower than _v12_ but higher than _v9_, it becomes a little troublesome, because 

1. In these versions nonlinear `FiniteElement` isn't implemented yet.

2. `Shooting` method can't handle the problem well, which is [an arguable backslide](https://mathematica.stackexchange.com/a/130373/1871).

[![enter image description here][1]][1]

_As we can see, though `ndsz` warning is generated, `NDSolve` manages to find the desired result in v9._

OK, so what to do? Well, to be honest I don't know if the following solution will cause other problem in _v10.0_, because _v10.0_ is a quite unstable version, but it does work in _v9_ and _v12.1_:
    
    rin = 10^-30;
    
    psol = ParametricNDSolveValue[{D[χ[x], {x, 2}] == (χ[x]^3/2)/Sqrt[x], χ'[
         10] == d, χ[10] == 0}, χ, {x, rin, 10}, d]
    
    drule = FindRoot[psol[d][rin] == 1, {d, 0}] // Quiet
    
    Plot[psol[d /. drule][r] // Evaluate, {r, rin, 10}]

  [1]: https://i.sstatic.net/eKZrT.png