It is well-known that $BU^+$ is homotopy equivalent to an infinite loop space where $U$ is the limit of the unitary groups $U(n)$ for $n \rightarrow \infty$ and $+$ denotes Quillen's Plus construction.
On the other hand there is a tool to detect infinite loop spaces by the action from $E\Sigma_p$ ($\Sigma_p$ is the symmetric group).
My question is: how does the map $E\Sigma_p \times_{\Sigma_p} BU^p \rightarrow BU$ looks like. Is it possible to write it down.
Firstly, no plus construction is needed here. The plus construction is used to kill a perfect subgroup of the fundamental group, but $\pi_1(BU)$ is already trivial.
Next, note that the space $\mathbb{C}^\infty=\bigoplus_{n=0}^\infty\mathbb{C}$ has a natural Hermitian inner product (with respect to which it is not complete). For any Hermitian space $\mathcal{V}$ that is isomorphic to $\mathbb{C}^\infty$, we consider the space $\mathcal{V}\oplus\mathcal{V}$ and its subspaces $\mathcal{V}_L=\mathcal{V}\oplus 0$ and $\mathcal{V}_R=0\oplus\mathcal{V}$. Let $B(\mathcal{V})$ denote the space of subspaces $\mathcal{A}\leq\mathcal{V}\oplus\mathcal{V}$ such that $\mathcal{A}\cap\mathcal{V}_L$ has finite codimension in $\mathcal{A}$, and also the same finite codimension in $\mathcal{V}_L$. To understand this in more detail, suppose we have a subspace $V\leq\mathcal{V}$ with $\dim(V)=n<\infty$, giving a decomposition $$ \mathcal{V}\oplus\mathcal{V} = V_L\oplus (V^\perp)_L \oplus V_R \oplus (V^\perp)_R. $$ We put $$ B(\mathcal{V};V) = \{A\oplus (V^\perp)_L : A\leq V_L\oplus V_R,\; \dim(W) = n\}. $$ We find that $B(\mathcal{V};V)$ is naturally identified with a finite-dimensional Grassmann manifold, so it has a natural compact Hausdorff topology. Moreover, the set $B(\mathcal{V})$ is the colimit of the sets $B(\mathcal{V};V)$, so we give it the colimit topology. One can check that $B(\mathcal{V})$ is then a model for the homotopy type $BU$.
Now suppose we have two Hermitian spaces $\mathcal{V}$ and $\mathcal{W}$ as above, and a linear map $\alpha\colon\mathcal{V}\to\mathcal{W}$ that preserves inner products. (This implies that $\alpha$ is injective, but it need not be surjective.) Given a point $\mathcal{A}=A\oplus(V^\perp)_L\in B(\mathcal{V};V)$, we have a point $$ \mathcal{B} = (\alpha\oplus\alpha)(A) \oplus(\alpha(V)^\perp)_R \in B(\mathcal{W};\alpha(V)). $$ One can check that this does not really depend on the choice of $V$, so we have a well-defined map $\alpha_*\colon B(\mathcal{V})\to B(\mathcal{W})$. One can also check that this is functorial. There are also evident maps $$ B(\mathcal{V})\times B(\mathcal{W}) \to B(\mathcal{V}\oplus\mathcal{W}), $$ making $B$ into a lax monoidal functor.
Now let $E(k)$ denote the space of inner-product preserving linear maps from $\mathcal{V}^k$ to $\mathcal{V}$. This has an evident action of $\Sigma_k$, which is free because inner-product preserving maps are always injective. It is a standard fact that $E(k)$ is also contractible, so it is a model for $E\Sigma_k$.
Now suppose we have elements $\mathcal{A}_1,\dotsc,\mathcal{A}_k\in B(\mathcal{V})$ and a map $\alpha\in E$. We then apply $\alpha_*$ to $\bigoplus_i\mathcal{A}_i$ to get a point $\gamma(\alpha;\mathcal{A}_1,\dotsc,\mathcal{A}_k)\in B(\mathcal{V})$. This construction gives the map $$ \gamma:E(k)\times_{\Sigma_k}B(\mathcal{V})^k\to B(\mathcal{V}) $$ that you need.
Note: An earlier version of this answer said that $\alpha_*(\mathcal{A})$ should just be $(\alpha\oplus\alpha)(\mathcal{A})$, but that is not correct, and in fact $(\alpha\oplus\alpha)(\mathcal{A})$ need not lie in $B(\mathcal{W})$. I thank Jack Smith for pointing out this error.
There are a number of equivalent ways of seeing $BU\times Z$ or $BU$ as an infinite loop space. The description that Neil gives is the action of the linear isometries operad $\mathcal{L}$ (complex version) on $BU$. A large number of related spaces have such a structure, as explained in the first section of the first chapter of "$E_{\infty}$ ring spaces and $E_{\infty}$ ring spectra", SLN 577 (1977). Morever, the Bott maps are maps of $\mathcal{L}$-spaces, which ties in Bott's original proof that $BU$ is an infinite loop space. The fact that $\coprod_n U(n)$ is a permutative category gives a quite different operad action on $\coprod_n BU(n)$, whose associated infinite loop space is $BU\times \mathbf{Z}$. It is not obvious a priori how to compare the infinite loop structures on these two models. The question is resolved in this and related examples (e.g. $BTop$) in my paper "The spectra associated to $\mathcal{I}$-monoids". Math. Proc. Camb. Phil. Soc. 84(1978), 313--322.
