We consider the Survivable Network Design problem (SNDP) in the single-pass insertion-only streaming model. The input to SNDP is an edge-weighted graph $G = (V, E)$ and an integer connectivity requirement $r(uv)$ for each $u, v \in V$. The objective is to find a min-weight subgraph $H \subseteq G$ s.t., for every pair of $u, v \in V$, $u$ and $v$ are $r(uv)$-edge/vertex-connected. Recent work by Jin et al. [JKMV24] obtained approximation algorithms for edge-connectivity augmentation, and via that, also derived algorithms for edge-connectivity SNDP (EC-SNDP). We consider vertex-connectivity setting (VC-SNDP) and obtain several results for it as well as improved results for EC-SNDP. * We provide a general framework for solving connectivity problems in streaming; this is based on a connection to fault-tolerant spanners. For VC-SNDP, we provide an $O(tk)$-approximation in $\tilde O(k^{1-1/t}n^{1 + 1/t})$ space, where $k$ is the maximum connectivity requirement, assuming an exact algorithm at the end of the stream. Using a refined LP-based analysis, we provide an $O(βt)$-approximation where $β$ is the integrality gap of the natural cut-based LP relaxation. When applied to the EC-SNDP, our framework provides an $O(t)$-approximation in $\tilde O(k^{1/2-1/(2t)}n^{1 + 1/t} + kn)$ space, improving the $O(t \log k)$-approximation of [JKMV24] using $\tilde O(kn^{1+1/t})$ space; this also extends to element-connectivity SNDP. * We consider vertex connectivity-augmentation in the link-arrival model. The input is a $k$-vertex-connected subgraph $G$, and the weighted links $L$ arrive in the stream; the goal is to store the min-weight set of links s.t. $G \cup L$ is $(k+1)$-vertex-connected. We obtain $O(1)$ approximations in near-linear space for $k = 1, 2$. Our result for $k=2$ is based on SPQR tree, a novel application for this well-known representation of $2$-connected graphs.