What is a non-ambiguous CFG for generating the set of natural numbers?
"From what I understand, there is not an algorithm to determine whether my guess is correct." It looks like your understanding is wrong.

There is no algorithm that determines whether any given formal language grammar is ambiguous context-free. There is no algorithm that determines whether any given formal language grammar is inherently ambiguous context-free. However, it is possible to decide many particular grammars is (inherently) ambiguous context-free or not by some algorithms.  For example, the trivial algorithm that just outputs "ambiguous context-free" determines the grammar with rules $S\to A|B$, $A\to c$, $B\to c$ as ambiguous context-free. For example, the trivial algorithm that always outputs "not inherently ambiguous context-free" determines all regular grammars as not inherently ambiguous context-free.

As said by rici, your grammar is regular. There is an algorithm that can decide any given regular grammar is ambiguous or not. (Immediately by definition, a regular language is context-free.). As demonstrated by Dmitri Urbanowicz, your grammar is, in fact, not ambiguous. 

As said by Dmitri Urbanowicz, your grammar is wrong since it cannot generate number 101. Here is a correct grammar, which can be shown to be non-ambiguous context-free.
$\quad S \to BT|A$
$\quad T \to AT|A$
$\quad A \to B|0$
$\quad B \to 1|2|3|4|5|6|7|8|9$