I am not sure if one can go up to $\Re(s)=1/2$ but if the local representation is tempered then one can certainly prove uniform continuity in your sense over compact sets in $\Re(s)>1/2$ ($1/2$ replaced by a bigger number for non-tempered representations). As we are considering continuity of the local zeta integral it is sufficient to pose the problem locally. Thus we think the Archimedean local component of $\pi$ is the principal series $\pi_\mu$ where $\mu\in \mathbb{C}$ such that the Laplacian eigenvalue of $f$ is $1/4-\mu^2$.

Let $\{\mathcal{D}_i\}$ be the set of invariant differential operators of degree $\le 2$. We define a Sobolev norm (semi-norm) on $\pi_\mu$ by $$S(v):=\sum_{i}\|\mathcal{D}_iv\|_\pi.$$ Thus to prove the uniform continuity it is enough to show the following: For every compact set $K\subseteq \{s\mid\Re(s)> 1/2\}$, $$|Z(W_{v_1},s)-Z(W_{v_2},s)|\ll_K S(v_1-v_2),$$ whenever $s\in K$. Using linearity of the zeta integral it is enough to show that $$Z(W_v,s)\ll_K S(v).$$ But the LHS is $$\int_{\mathbb{R}^\times}W_v(a(y))|y|^{s-1/2}d^\times y\ll_KS(v)\int_{\mathbb{R}^\times}\min(|y|^{1/2-\eta},|y|^{-2})|y|^{\Re(s)-1/2}\frac{dy}{|y|}.$$ We have used Proposition 3.2.3 of Michel-Venkatesh in a slightly modified form (their seminorm is only on the Kirillov model of $\pi_\mu$). If $\pi_\mu$ is tempered, i.e. $\mu$ is purely imaginary then $\eta$ can be replaced by any $\epsilon > 0$ (with $\ll_K$ replaced by $\ll_{K,\epsilon}$). As $\Re(s)> 1/2$, the last integral is $\ll_K 1$ choosing $\epsilon$ small enough.

As we are considering continuity of the local zeta integral it is sufficient to pose the problem locally. Thus we think the Archimedean local component of $\pi$ is the principal series $\pi_\mu$ where $\mu\in \mathbb{C}$ such that the Laplacian eigenvalue of $f$ is $1/4-\mu^2$.

Let $\{\mathcal{D}_i\}$ be the set of invariant differential operators of degree $\le 2$. We define a Sobolev norm (semi-norm) on $\pi_\mu$ by $$S(v):=\sum_{i}\|\mathcal{D}_iv\|_\pi.$$ Thus to prove the uniform continuity it is enough to show the following: For every compact set $K\subseteq \{s\mid\Re(s)> 0\}$, $$|Z(W_{v_1},s)-Z(W_{v_2},s)|\ll_K S(v_1-v_2),$$ whenever $s\in K$. Using linearity of the zeta integral it is enough to show that $$Z(W_v,s)\ll_K S(v).$$ But the LHS is $$\int_{\mathbb{R}^\times}W_v(a(y))|y|^{s-1/2}d^\times y\ll_KS(v)\int_{\mathbb{R}^\times}\min(|y|^{1/2-\eta},|y|^{-2})|y|^{\Re(s)-1/2}\frac{dy}{|y|}.$$ We have used Proposition 3.2.3 of Michel-Venkatesh in a slightly modified form (their seminorm is only on the Kirillov model of $\pi_\mu$). If $\pi_\mu$ is tempered, i.e. $\mu$ is purely imaginary then $\eta$ can be replaced by any $\epsilon > 0$ (with $\ll_K$ replaced by $\ll_{K,\epsilon}$). As $\Re(s)> 0$, the last integral is $\ll_K 1$ choosing $\epsilon$ small enough.