Is zero heat capacity possible without violating the third law of thermodynamics? Suppose we have a gapped system i.e. no gapless excitation is possible.  If the thermal energy is insufficient to excite atoms from ground state to excited state of any kind (of a single atom or of a collection of atoms) i.e. $k T \ll \Delta{E} = E_e - E_g$, can the system store any thermal energy?  If yes in what form is the thermal energy stored?
Alternative statement of my question: is zero heat capacity possible without violating the third law of thermodynamics i.e. at nonzero temperature?
 A: Mostly kinetic energy.
The kinetic energy of a free particle is not quantized. It becomes so when the particle is closed in a box. But even in this case the energy levels are often so closely spaced that the spectrum is almost continuous.
In fact, if you solve the Schroedinger equation for a particle in a 1D infinite square well you will find the following energy levels:
$$E_n = \frac{( \pi \hbar)^2}{2 m L^2} n^2 $$
where $L$ is the length of the box and $n=1,2,3,\dots$.
Let's put some numbers in the above formula. If $m$ is the mass of an hydrogen atom ($\sim 10^{-27}$ kg) and $L=1$cm, we will get
$$E_n \simeq (2.0 \cdot 10^{-18} \text{eV}\ ) \ n^2$$
So the difference in energy between the ground state ($n=1$) and the first excited state ($n=2$) is
$$E_2-E_1 = 3 \cdot (2.0 \cdot 10^{-18} \text{eV}\ ) = 6.0 \cdot 10^{-18} \text{eV}\  $$
At ambient temperature, $T\simeq300$K, we have
$$kT \simeq 2.6 \cdot 10^{-2} \text{eV}$$
You can see how this energy is enormous with respect to $E_2-E_1$:
$$\frac{kT}{E_2-E_1} = 4.33 \cdot 10^{15}$$
