I am trying to visualize that any vector bundle is a pullback from the Tautological bundle of Grassmannian.
If the tangent bundle of $S^2$ is not a simple example, is there a/the simplest example? The tangent bundle of torus $T^2$ is a trivial bundle so that doesn't count.
For vector bundles over manifolds there is a particularly simple construction, it goes as follows in your example. Embed $TS^2 \hookrightarrow \mathbf{R}^n \hookrightarrow\mathbf{R}^\infty$ (for instance by embedding $S^2 \hookrightarrow \mathbf{R}^k$ and getting $TS^2 \hookrightarrow \mathbf{R}^{2k}$) and map $x \in S^2$ to $T_xS^2 \in \mathrm{Gr}(2, n) \subset \mathrm{Gr}(2, \infty)$. Since different embeddings are isotopic (in $\mathbf{R}^\infty$), this map does not depend, upto homotopy, on the choice of embedding.
In the case of the tangent bundle, obtained by embedding the underlying manifold, this is often called the Gauss map.