Can I express the rate of a reaction as percent per second (%/s)?
According to IUPAC Recommendations for chemical kinetics [1], rates $\nu$ are defined for reactant $\ce{B}$ and product $\ce{Y}$ as a time $(t)$ derivative of the amount $n$ (in general) or per unit volume $V$ (for the closed system):

$$
\begin{array}{lll}
\hline
\text{System} &amp; \text{Consumption} &amp; \text{Formation} \\
\hline
\text{Open} &amp; \displaystyle\nu(n_\ce{B}) = -\frac{\mathrm dn_\ce{B}}{\mathrm dt} &amp; \displaystyle\nu(n_\ce{Y}) = \frac{\mathrm dn_\ce{Y}}{\mathrm dt} \\
\hline
\text{Closed} &amp; \displaystyle\nu(c_\ce{B}) = -\frac{1}{V}\frac{\mathrm dn_\ce{B}}{\mathrm dt} &amp; \displaystyle\nu(c_\ce{Y}) = \frac{1}{V}\frac{\mathrm dn_\ce{Y}}{\mathrm dt} \\
  &amp; \displaystyle\nu(c_\ce{B})|_{V = \mathrm{const}} = -\frac{\mathrm d[\ce{B}]}{\mathrm dt} &amp; \displaystyle\nu(c_\ce{Y})|_{V = \mathrm{const}} = \frac{\mathrm d[\ce{Y}]}{\mathrm dt} \\
\hline
\end{array}
$$

Rates of consumption and formation that are measured in $\pu{mol L-1 s-1}$ are defined for the closed system at constant volume, and molar concentration arises from its definition.
Whether you can or cannot substitute molar concentration $c_i$ with fractions can be answered by looking at the relationship between the two.

For example, molar concentration $c_i$ expressed in terms of mass fraction $\omega_i$ via the density of solution $\rho$ and the molar mass $M_i$ of the $i$th component:

$$c_i = \frac{\omega_i \rho}{M_i}\tag{1}$$

Similarly, molar concentration $c_i$ expressed in terms of mole fraction $x_i$ via the density of solution $\rho$ and the average molar mass $\bar{M}$ of the solution:

$$c_i = \frac{x_i\rho}{\bar{M}}\tag{2}$$

As you can see, there is a problem: as the reaction progresses, not only the amounts (or fractions) would change, but also the density and the molar mass of the solution may and will change in most cases, which leads to inapplicability of percent fractions as time derivatives.
All in all, in chemistry it's just easier to manipulate with the quantities based on amount of substance.

References


Laidler, K. J. A Glossary of Terms Used in Chemical Kinetics, Including Reaction Dynamics (IUPAC Recommendations 1996). Pure and Applied Chemistry 2009, 68 (1), 149–192. DOI: 10.1351/pac199668010149. (Free Access)