This is going to be a vague question, because I think I have an answer to a question I don't know how to state correctly. So bear with me for a bit, please!
Say we have a (smooth, if needed) complex toric variety $X$ with the corresponding fan $\Delta$ in $\mathbb{R}^n$. The choice of a basis fixes a coordinate system on the open orbit $U$, which is isomorphic to $(\mathbb{C}^\times)^n.$ In it there is a nice choice of generators $t_1,\ldots,t_n$ of $\pi_1(U,\ast)$ wrapping once counterclockwise around each coordinate hyperplane.
Take a ray $\sigma$ of $\Delta$. It corresponds to a torus-invariant divisors $D$. It seems that if we take an integral generating vector $v$ of the ray and write it in terms of our basis as $v=(v_{1},\ldots,v_n)$, then we get a choice of an element $\gamma$ of $\pi_1(U,\ast)$ "wrapping once around $D$". This is done by setting $\gamma=\prod\limits_{j=1}^n t_j^{v_j}.$
I want to understand what's going here without using coordinates. Is there a sense in which this construction is canonical? Or a way to make the words "wrapping once around $D$" precise?
Does Section 12.1 (on "The Fundamental Group" in "Topology of Toric Varieties") of Cox-Little-Schenck answer your question? There, they study the fundamental group of a toric variety in terms of torus orbits from the fan, as you describe.