What algorithms exist for solving natural number linear systems?
Your problem is NP-complete, by reduction from Subset Sum (it is in NP since the fact that everything is non-negative bounds the coefficients of the solution sufficiently well). Given an instance $S = \{s_1,\ldots,s_n\}, T$ of Subset Sum (is there a subset of $S$ summing to $T$?), we construct an instance $v_1,\ldots,v_{2n},u$ of your problem as follows. For each $1 \leq i \leq n$, we put $v_i$ to be the vector with two non-zero entries: $v_{i,i} = 1$ and $v_{i,n+1} = s_i$, and $v_{n+i}$ to be the vector with a unique non-zero entry $v_{n+i,i} = 1$. The target vector is $u=1,\ldots,1,T$. Each natural combination of $v_1,\ldots,v_{2n}$ equal to $1,\ldots,1,\ast$ must select exactly one of each of $v_i,v_{n+i}$, and so encodes a subset of $S$ whose sum is the value of the last component.