Can every multivariate distribution be expressed as a function of univariate distributions of the same random variables? Can every multivariate distribution $p(X)$ of a multivariate random variable $X = [X_1, X_2, \dots, X_d]^{T} \in \mathbb{R}^d$, be defined as some function of univariate distributions on $X_i$?
I cannot think of any reason this should be true, but I cannot find a counterexample.

YES
This is Sklar's theorem.
Let $H(x_1,\cdots,x_d) = \mathbb P\big(X_1 < x_1,\cdots,X_d<x_d\big)$ be a multivariate CDF, and let $F_i$ be the marginal CDFs. Then there is a function, $C$, called a copula, such that $H(x_1,\cdots,x_d) = C\big(F_{1}(x_1),\cdots,F_{d}(x_d)\big)$.
Depending on how gross the multivariate distribution is, it is possible to write this as a density, which I find more intuitive. In some regard, this is just the product of the margins times some function that governs (or describes) the dependence structure.
$$
h(x_1,\cdots,x_d) = \bigg(\prod f_i(x_i)\bigg)\times c\big(F_{1}(x_1),\cdots,F_{d}(x_d)\big)
$$
