Can we determine the amplitude of vibration of O-H bonds in water?
My question is: is it possible to know the amplitude of
  oscillation/stiffness constant of a O−H bond?


Yes.

Your approach is along the right lines but - 

The Hooke's Law formula you used

$$E = \frac{1}{2}kA^2$$

is for the stretching of a spring.  Hooke's law changes if the spring has mas(ses) attached at the end(s), now it is a harmonic oscillator.  In this case Hooke's Law becomes

$$E = \frac {h}{2 \pi} \sqrt{\frac{k}{\mu}}$$

where k is the force constant you are interested in and $\mathrm{\mu}$ is the reduced mass of the system. 

Since

$$E=h\nu$$

The frequency, which can be experimentally measured, is given as

$$\mathrm{frequency} = \frac {1}{2 \pi} \sqrt{\frac{k}{\mu}}$$

The frequency is independent of the amplitude, affected only by the masses and the stiffness of the spring.

If the masses of the atoms at the ends of the spring are given as A and B, then

$$\mu =\frac{A \cdot B}{A+B}$$

So for an $\ce{O-H}$ bond $\mathrm{\mu=\frac{1 \cdot 16}{1+16}}\ \mathrm{u}$. If we measure the frequency of the $\ce{O-H}$ stretch absorption, then we can calculate the force constant.

Here is a link to a one-page example that shows the actual computation and explains the specific units to be used very nicely.

You also ask about the amplitude of oscillation.  If, by this you mean the bond length, then yes this can also be calculated from spectra (relatively easy in simple diatiomic cases, but more difficult in complex polyatomic cases.  The methodology involves assuming a rigid rotor and then using the Schrodinger equation to solve for the energy states and extracting the bond length.  A nice example using $\ce{HCl}$ can be found here, look down the page for the section entitled "Bond Length of HCl".