After looking at the actual construction of the fiberwise embedding of a bundle $M \rightarrow E \rightarrow B$ into $B \times \mathbb{R}^N$, it became clear how to adjust it to embed the mapping cylinder. For convenience, let's assume that $B$ is a finite CW complex.

To obtain a fiberwise embedding of the bundle $p:E \rightarrow B$ into $B \times \mathbb{R}^N$, simply pick an embedding $i$ of $E$ into $\mathbb{R}^N$ and define $E \rightarrow B \times \mathbb{R}^N$ by $x \rightarrow (p(x),i(x))$.

Suppose now we have a fiberwise homotopy equivalence between $E \rightarrow B$ and $E' \rightarrow B'$. Let $I$ be a relative embedding of the mapping cylinder into $\mathbb{R}^L \times (0,1]$. Now we can define a fiberwise embedding of the mapping cylinder of the bundle equivalence by $p \rightarrow (p(x),I(x))$.

After looking at the actual construction of the fiberwise embedding of a bundle $M \rightarrow E \rightarrow B$ into $B \times \mathbb{R}^N$, it became clear how to adjust it to embed the mapping cylinder. For convenience, let's assume that $B$ is a finite CW complex.

To obtain a fiberwise embedding of the bundle $p:E \rightarrow B$ into $B \times \mathbb{R}^N$, simply pick an embedding $i$ of $E$ into $\mathbb{R}^N$ and define $E \rightarrow B \times \mathbb{R}^N$ by $x \rightarrow (p(x),i(x))$.

Suppose now we have a fiberwise homotopy equivalence between $E \rightarrow B$ and $E' \rightarrow B'$. Let $I$ be a relative embedding of the mapping cylinder into $\mathbb{R}^L \times (0,1]$. Now we can define a fiberwise embedding of the mapping cylinder of the bundle equivalence by $x \rightarrow (p(x),I(x))$.