The symmetric convex hull of random points that are independent and distributed according to the cone probability measure on the $\ell_p$-unit sphere of $\mathbb R^n$ for some $1\leq p < \infty$ is considered. We prove that these random polytopes have uniformly absolutely bounded isotropic constants with overwhelming probability. This generalizes the result for the Euclidean sphere ($p=2$) obtained by D. Alonso-Gutiérrez. The proof requires several different tools including a probabilistic representation of the cone measure due to G. Schechtman and J. Zinn and moment estimates for sums of independent random variables with log-concave tails originating in the work of E. Gluskin and S. Kwapień.