For the symmetric group $\Sigma_n$, you can take $$ \begin{align*} E\Sigma_n = &= \{\text{injective functions } \{1,\dotsc,n\}\to\mathbb{R}^\infty \}$$ $$ } \\ B\Sigma_n = &= \{\text{subsets of size $n$ in } \mathbb{R}^\infty \}$$} \end{align*}

Now let $G_n$ be the group of braids on $n$ strings, and let $H_n$ be the subgroup of pure braids. We have \begin{align*} BH_n &= \{\text{injective functions } \{1,\dotsc,n\}\to\mathbb{R}^2 \} \\ BG_n &= \{\text{subsets of size $n$ in } \mathbb{R}^2 \} \end{align*} These spaces have trivial homotopy groups $\pi_{k}(X)$ for $k\geq 2$, so $$ EH_n=EG_n= \text{ universal cover of } BH_n = \text{ universal cover of } EH_n. $$ I think I see a proof that this space is homeomorphic to $\mathbb{R}^{2n}$, but I don't know if that is in the literature.

For the symmetric group $\Sigma_n$, you can take $$ E\Sigma_n = \{\text{injective functions} \{1,\dotsc,n\}\to\mathbb{R}^\infty \}$$ $$ B\Sigma_n = \{\text{subsets of size $n$ in } \mathbb{R}^\infty \}$$