To solve this problem, we need to understand the concept of a buffer solution and the Henderson-Hasselbalch equation.

1. Understanding Buffer Solutions:
A buffer solution is a solution that resists changes in pH when small amounts of acid or base are added to it. It consists of a weak acid and its conjugate base or a weak base and its conjugate acid.

2. Henderson-Hasselbalch Equation:
The Henderson-Hasselbalch equation relates the pH of a buffer solution to the pKa of the weak acid and the concentration of the acid and its conjugate base.
pH = pKa + log([A-]/[HA])
Where:
pH is the desired pH of the buffer solution
pKa is the acid dissociation constant of the weak acid
[A-] is the concentration of the conjugate base
[HA] is the concentration of the weak acid

3. Given Information:
- pKa of NH4OH (weak base) = 4.7
- pH of the desired buffer solution = 8.7
- Concentration of NH4OH = 0.1 M

4. Calculating the Concentration of NH4+ and NH3:
Since NH4OH is a weak base, it will dissociate into NH4+ (conjugate acid) and OH- ions in water. However, in an acidic solution, NH4+ will further react with H+ ions to form NH3 (conjugate base) and H2O. Thus, the concentration of NH4+ and NH3 will depend on the pH of the solution.

Using the Henderson-Hasselbalch equation, we can rearrange it as follows:
pH = pKa + log([A-]/[HA])
8.7 = 4.7 + log([NH3]/[NH4+])

Taking the antilog of both sides, we have:
[NH3]/[NH4+] = 10^(8.7-4.7) = 10^4 = 10,000

5. Calculating the Volume of NH4Cl Solution:
We know that the concentration of NH4OH is 0.1 M, and we need to find the volume of 0.2 M NH4Cl solution that should be added to 100 ml of NH4OH solution to produce the desired buffer solution.

Let the volume of NH4Cl solution required be V ml.
The moles of NH4OH in 100 ml = 0.1 x 0.1 = 0.01 moles
The moles of NH4Cl in V ml = 0.2 x V/100

Since NH4Cl and NH4OH react in a 1:1 ratio, the moles of NH4Cl and NH4OH should be equal in the buffer solution.
0.01 = 0.2 x V/100
V = (0.01 x 100)/0.2 = 5 ml

6. Answer:
Therefore, the volume of 0.2 M NH4Cl solution that should be added to 100 ml of 0.1 M NH4OH solution to produce a buffer solution of pH = 8.7 is 200 ml (option C).

C is correct.