While the actual question has been answered very quickly in the comments, there are some interesting results in the higher-dimensional case, concerned with finitely supported functions $f : \mathbb{Z}^n \to \mathbb{R}$. I will mainly address the case of *nonnegative* Laplace transform after discussing the strictly positive case. It may be possible to both of these cases via Pólya's Positivstellensatz, using a small perturbation in the case of nonnegative Laplace transform, but I'm not sure at this point.

In both cases, the question is secretly about Laurent polynomials, as I will explain now in more detail. Readers who understand this can skip to the headings below.

The finitely supported functions $f : \mathbb{Z}^n \to \mathbb{R}$ form a ring under pointwise addition and convolution as multiplication. Equivalently, it may be convenient to consider it as the ring of finitely supported signed measures on $\mathbb{Z}^n$ rather than functions. As pointed out by David Handelman in the comments, this ring is also isomorphic to the ring of Laurent polynomials $\mathbb{R}[X_1^\pm,\ldots,X_n^\pm]$, where the generator $X_i$ corresponds to the Dirac measure $\delta_{e_i}$. Upon equipping this ring with the coefficientwise order, or equivalently with the pointwise order on functions, we get an ordered commutative ring in the sense of a ring equipped with a subset of positive elements $P$ which is closed under addition and multiplication. Hence we are effectively dealing with a problem in real algebraic geometry.

The monotone ring homomorphisms $\mathbb{R}[X_1^\pm,\ldots,X_n^\pm]$ are precisely the evaluation maps at points $s\in\mathbb{R}^n_{> 0}$. Writing $s$ as a componentwise exponential, $s_i = e^{t_i}$ for $t\in\mathbb{R}^n$, makes the connection with the Laplace transform: the monotone homomorphisms from functions to the reals are parametrized by $t\in\mathbb{R}^n$, and are given by the values of the Laplace transform
$$f \longmapsto \sum_{k\in\mathbb{Z}^n} f(k)\, e^{\langle t,k\rangle}.$$

**Strictly positive Laplace transform**

In this case, we have (see also discussion in the comments):

**Theorem.** If finitely supported $f : \mathbb{Z}^n \to \mathbb{R}$ has strictly normalized Laplace transform
$$ \frac{\sum_k f(k) e^{\langle t,k\rangle}}{\sum_k |f(k)| e^{\langle t,k\rangle}},$$
for all nonzero $t\in(\mathbb{R}\cup\{-\infty\})^n$, then there is $m\in\mathbb{N}$ such that $\left(1 + \sum_i \delta_{e_i}\right) ^{\ast m}\ast f$ is nonnegative.

*Proof:* By suitable translation, we can assume that $f$ is supported on $\mathbb{Z}^n_+$. Then we can formulate it in terms of polynomials, in which case the homogenized form is the most natural: if $p$ is a homogeneous polynomial whihc is strictly positive on the closed simplex, then there is $k\in\mathbb{N}$ such that $\left(\sum_i X_i\right)^k f$ has nonnegative coefficients. This is Pólya's Positivstellensatz (which also exists in a version in which all coefficients are strictly positive).

As explained in David Handelman's answer, one can also formulate a version of this result which gives a necessary and sufficient condition.

**Nonnegative Laplace transform**

In this case, we can only expect a $g$ as in the OP to exist *approximately*. The following is Theorem 5.9(a)-(b) of this recent paper.

**Theorem.** For finitely supported $f : \mathbb{Z}^n\to\mathbb{R}$, the following are equivalent:

- The Laplace transform of $f$ is nonnegative.
- For every $\varepsilon > 0$ and $r\in\mathbb{R}_+$, there exist a finitely supported nonzero and nonnegative $g : \mathbb{Z}^n \to \mathbb{R}$ and a polynomial $p\in\mathbb{R}_+[X]$ such that $p(r) \leq \varepsilon$ and the function
$$g \ast \left[f + p\left(1 + \sum_i (\delta_{e_i} + \delta_{-e_i}) \right)\right]$$
is nonnegative.

Here, the polynomial $p\left(1 + \sum_i (\delta_{e_i} + \delta_{-e_i}) \right)$ is to be understood with respect to convolution as multiplication. It represents a correction whose Laplace transform converges to zero pointwise in the limit $\varepsilon\to 0, \: r\to\infty$.

There are many other statements of a similar flavour, some of which can be deduced from the existing results of the above paper and some of which are part of work in progress.