Many results of this type can be proven by a basic existence of rational points argument. For any finite set of nontrivial linear equations, we can find a set of integers that does not satisfy any of them. To replace the infinite set of equations in the question with a finite set, we need some kind of bound on the $a_i$, which we obtain from a bound on the $x_i$.

If $N< x_1, \dots, x_k < 2N$, then any $a_i$ satisfying the equation $\sum _{i=0}^k a_i x_i = \sum_{i=0}^k 2 x_i$ must satisfy $\sum_{i=0}^k a_i \leq 4 k$. This gives us finitely many linear equations to not satisfy. Specifically, the number of nonnegative integer solutions to $\sum_{i=0}^k a_i \leq 4k$ is $\left(\begin{array}{c} 5k \\ k \end{array}\right)$ because it is equal to the number of ways to place $4k$ balls into $k+1$ boxes.

It takes $N-1$ linear equations to cover all the integer points in the hypercube $N< x_1, \dots, x_k < 2N$, since there are $(N-1)^k$ points and each hyperplane can contain at most $(N-1)^{k-1}$. So setting $N= \left(\begin{array}{c} 5k \\ k \end{array}\right)+2$, we can find $x_1,\dots,x_k$ which have your desired property.

This bound is exponential, about $12.21^k$. One could be much more careful in finding the bound by choosing the convex region more carefully, eliminating more of the equations, using better estimates for the number of points on the hyperplanes, and so on. I'm not sure how much one could improve this bound - could one do better than exponential?