If we take as base ring $\mathbb{Q}$ in place of $\mathbb{Z}$, the answer is Yes.
You surely have good reasons for taking the integers as base ring. But in view of your more general question Minimal rank of a permutation resolution of a $G$-lattice I believe that over $\mathbb{Q}$ (any other field those char. doesn't divide the group order will also do) there is a fair chance to solve the general question. Therefore the subsequent consideration might be of interest to you.
Let $G = C_p \times C_p= \langle \sigma,\tau\rangle$ and for $H \le G$ let $e_H = \sum_{h \in H} h \in \mathbb{Q}[G]$. Define a map of $\mathbb{Q}[G]$-modules by
$$\phi_H: \mathbb{Q}[G/H] \to \mathbb{Q}[G],\,\,gH \mapsto g\cdot e_H.$$
Let $H_j = \langle \sigma\tau^j\rangle\,\,(j=0,...,p-1)$ and $H_p=\langle\tau\rangle$ and
$$\phi: \bigoplus_{i=1}^{p-1}\mathbb{Q}[G/H_j] \to \mathbb{Q}[G]\twoheadrightarrow L.$$
Note: The left hand side is a $\mathbb{Q}$-vector space of dimension $(p-1)p$.
Claim: $\phi$ is an epimorphism.
Since $H_i \cap H_j = 1$, we find $\sum_{j=0}^p e_{H_j}= p + e_G$. Moreover, $L = \mathbb{Q}[G]/\langle e_{H_0},e_{H_p}\rangle$ and $e_G = e_{H_0}e_{H_p}$. Hence $e_{H_0},e_{H_p},e_G$ vanish in $L$. Therefore $\phi(\sum_{j=1}^{p-1}H_j) = p \in L$ and $\phi(\frac{1}{p} \sum_{j=1}^{p-1}gH_j) = \bar{g} \in L$ for all $g \in G$. This shows the surjectivity of $\phi$.
