In the solution to this problem, why is the electron charge omitted?
Yes, they should include the basic charge e. But they have gotten the correct numerical answer, because when they substituted for \$kT\$ they used the value for \$\frac{kT}{q}\$.
That is,
$$\frac{kT}{q}\approx 0.026\ {\rm V}$$
For solving problems of this type, it's easier to just remember this value as the &quot;thermal voltage&quot;, often designated \$V_T\$. Then the Shockley diode equation becomes
$$I = I_s \left[\exp\left( \frac{V}{V_T}\right)  - 1\right].$$
(Also notice, in your solution, you moved the -1 term inside the exponential function, which is incorrect)
So, while they dropped the \$q\$ from their expression, they actually used it when they calculated the numerical value.
In your solution, you just accepted their value for \$kT\$ and thus got an incorrect final result. If you want to account for \$q\$ separately, you should use
$$kT=(1.38\times10^{-23}\ {\rm J\cdot K^{-1}})(300\ {\rm K})=4.14\times 10^{-21}\ {\rm J}.$$