We study the limiting spectral distribution of large-dimensional sample covariance matrices associated with the order-d symmetric random tensors formed by products of d variables chosen from n independent standardized random variables. We find optimal sufficient conditions for this distribution to be the Marchenko-Pastur law in the case $d=d(n)$ and $n\to \mathrm{\infty }$. Our conditions reduce to $d^2=o(n)$ when the variables have uniformly bounded fourth moments. The proofs are based on a new concentration inequality for quadratic forms in symmetric random tensors and a law of large numbers for elementary symmetric random polynomials.