Any way to determine if a finite multiset of natural numbers can be combined via addition or subtraction to form zero?

This is an NP-complete problem, equivalent to the Subset-sum problem. If $T$ is the sum of all your numbers $x_n$, your problem is equivalent to finding a subset whose sum is $T/2$. Given such a subset, you give the members of that subset the sign $-$ and the others the sign $+$.

Thus there is no known polynomial-time algorithm, but there is a pseudo-polynomial time dynamical programming algorithm. It goes like this. Suppose your numbers are $x_1, \ldots, x_n$ (assumed to be positive integers). For integers $1 \le m \le n$ and $0 \le t \le T/2$, let $I(m,t) = 1$ if there is a subset of $x_1, \ldots, x_m$ whose sum is $t$, and $0$ if not. You compute it as follows. Start with $I(1,0) = 1$, $I(1,x_1) = 1$, all others $0$. Then given the $I(m,t)$, $I(m+1,t) = 1$ if either $I(m,t) = 1$ or ($t \ge x_{m+1}$ and $I(m,t-x_{m+1}) = 1$). The answer to your problem is yes if and only if $I(n,T/2) = 1$ (of course $T/2$ must be on integer, so $T$ must be even).