What is equivalent to the Axiom of Choice is the assertion that every cartesian product of any nonempty family of nonempty sets is itself nonempty (not merely that "an infinite Cartesian product of a non-empty family of nonempty sets").
There are plenty of families for which one can prove, without needing to invoke the Axiom of Choice, that their cartesian product is nonempty. For example, as you note, a product of any nonempty family of nonempty well-ordered sets is nonempty, regardless of the cardinality of the family. (However, proving that a denumerably infinite product of denumerably infinite sets is nonempty requires at least some Choice; you can think of this as a weakening of the previous case, since here we are saying that the sets are well-order able but not necessarily well-order ed. That is, rather than coming already provided with a well-ordering, we are just told that a well ordering exists).
In general, any family in which all the sets are the same also has nonempty product: if $\{A_i\}_{i\in I}$ is a nonempty family, and $A_i=A_j\neq\emptyset$ for all $i,j\in I$, then since $A_i$ is nonempty, there exists $a\in A_i$. Then the function $f\colon I\to\cup A_i$ given by $f(i) = a$ for all $i\in I$ is a choice function for the family (and an element of $\times_{i\in I} A_i$). In particular, both the sets $\mathbb{R}^\mathbb{R}$ and $\mathbb{R}^{\mathbb{N}}$ are nonempty, and we can prove this without invoking choice. Just the $f\colon\mathbb{R}\to\mathbb{R}$ given by $f(r)=0$ for all $r$; this is an element of $\mathbb{R}^{\mathbb{R}}$; and $g(n)=0$ for all $n\in\mathbb{N}$, this is an element of $\mathbb{R}^{\mathbb{N}}$. Neither one requires AC.
Likewise, any family $\{A_i\}_{i\in I}$ in which there is a cofinite $J\subseteq I$ with $\cap_{j\in J}A_j\neq\emptyset$ will have a choice function whose existence does not depend on AC: take any $J$ with the given property, take any $x\in \cap_{j\in J}A_j$, and letting $I-J = \{i_1,\ldots,i_k\}$, pick $a_t\in A_{i_t}$, $t=1,\ldots,k$. Then $f\colon J\to\cup A_i$ given by $f(i) = \left\{\begin{array}{ll} x & \mbox{if $i\in J$;}\\ a_1 &\mbox{if $i=i_1$;}\\ \vdots\\ a_k &\mbox{if $i=i_k$.} \end{array}\right.$ is a choice function for the family, whose existence can be established without invoking the Axiom of Choice. This is of course not necessary, merely sufficient.