Following up on the answer of Simon Henry, let us prove the following statement. For a pro-space $\hat{X} = \{X_i\}_{i \in I}$, we let $Spaces_{/\hat{X}}$ denote the $\infty$-topos defined as the (cofiltered) limit in $Topoi$ of the $I$-family of étale topoi $Spaces_{/X_i}$. We will refer to such $\infty$-topoi as pro-étale $\infty$-topoi.
Claim: Suppose that ${\cal X}$ is an $\infty$-topos which is a loop object in $Topoi$. Then ${\cal X}$ is a left exact localization of a pro-étale $\infty$-topos $Spaces_{/\hat{X}}$ for $\hat{X} \in Pro(Spaces)$. If ${\cal X}$ is a double loop object then ${\cal X}$ itself is a pro-étale $\infty$-topos. In particular, every spectrum object in $Topoi$ consists of pro-étale $\infty$-topoi.
Proof:
Let ${\cal Y}$ be an $\infty$-topos equipped with two points $x_*,y_*: Spaces \to {\cal Y}$. Before considering the associated limit $Spaces \times_{\cal Y} Spaces$ we can consider the corresponding lax limit (or comma object) $Spaces \times^{\rm lax}_{\cal Y} Spaces$. We claim that this comma object exists and is furthermore a pro-étale $\infty$-topos. Indeed, let ${\cal Z}$ be an $\infty$-topos and let $p_*: {\cal Z} \to Spaces$ denote the terminal map. Then the data of a natural transformation $x_*p_* \Rightarrow y_*p_*$ is equivalent, by adjunction, to the data of a natural transformation $y^*x_* \Rightarrow p_*p^*$ of functors from $Spaces$ to $Spaces$. We note that both $y^*x_*$ and $p_*p^*$ are left exact functors and are hence corepresentable by pro-spaces, where the pro-space $Shp({\cal Z})$ corepresnting $p_*p^*$ is also known as the shape of ${\cal Z}$. Let $\hat{P}_{x,y} \in Pro(Spaces)$ denote the pro-space corepresenting $y^*x_*$. We then get that the data of a natural transformation $x_*p_* \Rightarrow y_*p_*$ is equivalent to the data of a map of pro-spaces $Shp({\cal Z}) \to \hat{P}_{x,y}$. We now recall that the formation of shapes ${\cal Z} \mapsto Shp({\cal Z})$ is left adjoint to the functor $\hat{X} \mapsto Spaces_{/\hat{X}}$ from pro-spaces to $\infty$-topoi. We may hence conclude that the data of a natural transformation $x_*p_* \Rightarrow y_*p_*$ is equivalent to the data of a geometric morphism ${\cal Z} \to Spaces_{/\hat{P}_{x,y}}$. We may then conclude that, if we let $q_*: Spaces_{/\hat{P}_{x,y}} \to Spaces$ be the terminal geometric morphism, then we have a canonical natural transformation $\tau:x_*q_* \Rightarrow y_*q_*$ which exhibits $Spaces_{/\hat{P}_{x,y}}$ as the desired comma object $Spaces \times^{\rm lax}_{\cal Y} Spaces$. Now let ${\cal P}_{x,y} \subseteq Spaces_{/\hat{P}_{x,y}}$ be the maximal left exact localization (see HTT 6.2.1.2) of $Spaces_{/\hat{P}_{x,y}}$ contained in the reflexive accessible subcategory
$$\{X \in Spaces_{/\hat{P}_{x,y}} | \tau_X:x_*q_*X \to y_*q_*X \text{ is an equivalence}\} \subseteq Spaces_{/\hat{P}_{x,y}}.$$
Comparing universal properties we see that ${\cal P}_{x,y} \simeq Spaces \times_{\cal Y} Spaces$ represents the corresponding limit. In particular, for every points $x_*: Spaces \to {\cal Y}$ the loop $\infty$-topos ${\cal P}_{x,x} \simeq \Omega_x{\cal Y}$ is a left exact localization of a pro-étale $\infty$-topos.
Now suppose that ${\cal X}$ is an $\infty$-topos which is a double loop object, i.e., ${\cal X} \simeq \Omega_x{\cal Y}$ where ${\cal Y}$ itself is a loop object in $Topoi$. By the above we then have that ${\cal Y}$ is a left exact localization of pro-étale $\infty$-topos $Spaces_{/\hat{Y}}$, for some pro-space $\hat{Y} = \{Y_i\}_{i \in I} \in Pro(Spaces)$. Then $Spaces_{/\hat{Y}} = \lim_i Spaces_{/Y_i}$ and hence the space of points $y_*: Spaces \to Spaces_{/\hat{Y}}$ is naturally equivalent to the space $\lim_i Y_i = {\rm Map}_{Pro(Spaces)}(\ast,\hat{Y}) \in Spaces$. In this case, if $y_*: Spaces \to Spaces_{/\hat{Y}}$ corresponds to a compatible collection of points $y_i \in Y_i$ then
$$ \Omega_{y}Spaces_{/\hat{Y}} = \Omega_{y}\lim_i Spaces_{/Y_i} \simeq \lim_i \Omega_{y_i} Spaces_{/Y_i} \simeq \lim_i Spaces_{/\Omega_{y_i} Y_i} = Spaces_{/\Omega_{y}\hat{Y}} .$$
Furthermore, if $y_*: Spaces \to Spaces_{/\hat{Y}}$ is a point which factors as $Spaces \stackrel{x_*}{\to} {\cal Y} \hookrightarrow Spaces_{/\hat{Y}}$ then $\Omega_y (Spaces_{/\hat{Y}}) \simeq \Omega_x {\cal Y}$. It then follows that ${\cal X} \simeq \Omega_x {\cal Y} \simeq Spaces_{/\Omega_y\hat{Y}}$ is a pro-étale $\infty$-topos, as desired.
$\Box$
Remarks:
1) At this point one may be tempted to conclude that every spectrum object in $Topoi$ is the image of a spectrum object in $Pro(Spaces)$. This is very possibly the case, but a-priori it does not follow from the above claim. All that can be deduced is that if we denote by $\widehat{Etale} \subseteq Topoi$ the full subcategory spanned by pro-étale topoi, i.e., the essential image of $Pro(Spaces) \to Topoi$, then every spectrum object in $Topoi$ comes from a spectrum object in $\widehat{Etale}$. However, since the functor $Pro(Spaces) \to \widehat{Etale}$ is not fully-faithful it is not a-priori clear if the map $Sp(Pro(Spaces)) \to Sp(\widehat{Etale})$ is essentially surjective. In other words, there could, in principle, be a spectrum object ${\cal X}_0,{\cal X}_1,...$ in $Topoi$ in which every ${\cal X}_i$ is a pro-étale spectrum ${\cal X}_1 \simeq Spaces_{/\hat{X}_i}$ but the structure equivalences $\varphi_i:{\cal X}_i \stackrel{\simeq}{\to} \Omega{\cal X}_{i+1}$ do not come from equivalences of pro-spaces $f_i:\hat{X}_i \stackrel{\simeq}{\to} \Omega\hat{X}_{i+1}$ (up to finitely many $\varphi_i$'s we can always arange it up to equivalence, but it's not clear if we can arrange all the $\varphi_i$'s at once; there is an obstruction to this which lies in a suitable $\lim^1$ set).
2) The claim that $Sh(S^1)$ is not an infinite loop object in $Topoi$ can be deduced from the above claim, at least if we assume that the truncation functor $\tau_{\leq 0}: Topoi \to Topoi_0$ from $\infty$-topoi to $0$-topoi (i.e., locales) preserves cofiltered limits (it seems to me that this claim should be deducible from the fact that cofiltered limits on both cases are computed in ${\rm Cat}_\infty$, see HTT 6.3.3.1). Assuming this, suppose that we had a CW complex $X$ such that $Sh(X)$ was an infinite loop object in $Topoi$. By the claim we have that $Sh(X) \simeq Spaces_{/\hat{X}} = \lim_i Spaces_{/X_i}$ for some pro-space $\hat{X} = \{X_i\}_{i \in I}$. By the commutativity of cofiltered limits and truncations we then have that the local $O(X)$ is equivalent to the local $\lim_i \tau_{\leq 0}(Spaces_{/X_i}) = \lim_i{\rm Sub}(\pi_0(X_i))$, where ${\rm Sub}(\pi_0(X_i))$ is the locale of subsets of $\pi_0(X_i)$. Since $X$ is Hausdorff it is sober and hence we may deduce that $X$ is homeomorphic to the limit $\lim_i \pi_0(X_i)$ (computed in topological spaces). But $X$ is a CW-complex and hence locally connected, and so $X \cong \lim_i \pi_0(X_i)$ would have to be a discrete set. In particular, $Sh(S^1)$ is not an infinite loop object (or even a double loop object).
