For any $ G$, you can always embed the affine Grassmannian $Gr_G$ into the space of $\mathbb C [[t]]$-lattices in $ \mathfrak{g} \otimes \mathbb C ((t))$. I'm pretty sure that this appears in Lusztig's paper.

Alternatively, for any $ G $, there is an embedding $ Gr_G \rightarrow \prod_{\lambda} \{ \text{lattices in } V(\lambda) \otimes \mathbb C ((t)) \}$. The image of
this embedding will be those sequences of lattices which are compatible with morphisms $ V(\lambda) \otimes V(\mu) \rightarrow V(\nu) $. See section 10.3 in Finkelberg-Mirkovic for more details.

Unfortunately, neither of these embeddings are as helpful as the one for $ GL_n$.