This is my suggested solution with a few details left out (thanks also go to Jonny Evans).

Choose an auxiliary Riemannian metric on $N$. Let $f\colon M \to N$ be a fixed isosymplectic embedding, i.e. such that $f^{*}\sigma = \omega$, and let $\pi\colon N_{f}M \to N$ denote the normal bundle of $f$ with respect to the metric on $N$. By the tubular neighbourhood theorem there is a diffeomorphism $$\nu_{f}\colon \mathcal{O}_{f} \subseteq N \to \mathcal{O}(0_{M}) \subseteq N_{f}M$$ from an open neighbourhood $\mathcal{O}_{f}$ of the image $\mathrm{im}\,f \subseteq N$ to an open neighbourhood $\mathcal{O}(0_{M})$ of the zero section of $N_{f}M$.

Let $g\colon M \to N$ be another isosymplectic embedding such that $\mathrm{im}\,g \subseteq \mathcal{O}_{f}$ and suppose that $g$ is $C^{1}$-close enough to $f$ so that the composition $\pi\circ \nu_{f} \circ g\colon M \to M$ is a diffeomorphism of $M$ (recall that $\mathrm{Diff}(M)$ is $C^{1}$-open in $C^{\infty}(M, M)$). Then $$s_{g} := \nu_{f}\circ g\circ (\pi \circ \nu_{f} \circ g)^{-1}\colon M \to N_{f}M$$ is a section of the vector bundle $\pi\colon N_{f}M \to M$.

Furthermore, denote by $\Gamma(N_{f}M)$ the space of (smooth) sections of $\pi$ and let $V \subseteq \Gamma(N_{f}M)$ be a $C^{1}$-small neighbourhood of the zero section. Define $$ U_{f} = \{g \in \mathcal{E} \,|\, \mathrm{im}\,g \subseteq \mathcal{O}_{f}, \pi \circ \nu_{f} \circ g \in \mathrm{Diff}(M), s_{g} \in V\}$$ and $$V_{f} = \{ s \in \Gamma(N_{f}M) \,|\, \mathrm{im}\,s \subseteq \mathcal{O}(0_{M}), s \in V\}.$$

The set $V_{f}$ is open in $\Gamma(N_{f}M)$ (in the compact-open $C^{\infty}$-toplogy) and so we could consider the 'chart' $$ U_{f} \ni g \mapsto (s_{g}, \pi \circ \nu_{f} \circ g) \in V_{f} \times \mathrm{Diff}(M) $$ but we would like to adjust the second factor so that it lands in $\mathrm{Symp}(M, \omega)$ instead. We will use Moser stability to achieve this.

Let $\kappa_{t}\colon N_{f}M \to N_{f}(M), t\in [0,1]$, be the fibre-preerving smooth homotopy defined by 'sliding along the fibres to the zero section,' i.e. $\kappa_{0} = \mathrm{id}$ and $\kappa_{1} = \text{projection onto the zero section}$.

For $s\in V_{f}$ arbitrary, $t\mapsto \kappa_{t}\circ s$ is a smooth homotopy of sections from $s$ to the zero section and, moreover, $$t \mapsto \omega^{s}_{t} := (\nu_{f}^{-1} \circ \kappa_{t}\circ s)^{*} \sigma $$ is a smooth homotopy of (cohomologous) symplectic forms on $M$ (if the neighbourhood $V$ above was chosen $C^{1}$-small enough) from $\omega_{0}^{s}$ to $\omega_{1}^{s} = f^{*}\sigma = \omega$ (*yes, this is the original $\omega$ on $M$*). By the Moser stability theorem there exists a smooth isotopy $h_{t}^{s} \in \mathrm{Diff}(M), t\in [0,1],$ such that $(h_{1}^{s})^{*}\omega = \omega_{0}^{s}$. Then $$ g_{s} := \nu_{f}^{-1} \circ s \circ (h_{1}^{s})^{-1}\colon M \to N$$ is an embedding such that $g_{s}^{*}\sigma = \omega$, hence $g_{s} \in \mathcal{E}$. In fact, $g_{s} \in U_{f}$.

Now we can define $$\varphi_{f}\colon V_{f}\times \mathrm{Symp}(M, \omega) \to U_{f}, \quad \varphi_{f}(s, a) = g_{s} \circ a = \nu_{f}^{-1}\circ s \circ (h_{1}^{s})^{-1} \circ a.$$ This is a bijection with the inverse $$\psi_{f}\colon U_{f} \to V_{f} \times \mathrm{Symp}(M, \omega), \quad \psi_{f}(g) = (s_{g}, h_{1}^{{s}_{g}}\circ \pi \circ \nu_{f} \circ g).$$

Since $\mathrm{Symp}(M, \omega)$ is a smooth manifold (for example in the convenient calculus setting of Frölicher, Kriegl and Michor), we can proclaim $\psi_{f}$ a local chart on $\mathcal{E}$. It is not difficult to check that if another $k\in \mathcal{E}$ is given, the transition map $$\psi_{k} \circ \psi_{f}^{-1} = \psi_{k} \circ \varphi_{f}\colon V_{f}\times \mathrm{Symp}(M, \omega) \to V_{k}\times \mathrm{Symp}(M, \omega)$$ is smooth (in the convenient calculus setting) -- the Moser stability term $h_{1}^{s}$ can be constructed canonically using Hodge theory, this way it will be in particular smoothly dependent on the section $s$.

The collection $(U_{f}, \varphi_{f})_{f \in \mathcal{E}}$ then defines a smooth atlas on the set $\mathcal{E}$. The topology on $\mathcal{E}$ induced by charts from the compact-open $C^{\infty}$-topology on $\Gamma(N_{f}M)$ and $\mathrm{Symp}(M, \omega)$ turns out to be Hausdorff and so $\mathcal{E}$ is a convenient smooth manifold.

Finally, for any $g\in U_{f}$ and $a\in \mathrm{Symp}(M, \omega)$ we have $s_{g\circ a} = s_{g}$ and so $$\psi_{f}(g\circ a) = \psi_{f}(g)\cdot a.$$ In other words, the chart $\psi_{f}$ descends to a chart on the quotient $\mathcal{B}$ with values in $V_{f} \subseteq \Gamma(N_{f}M)$ thus defining a smooth structure on $\mathcal{B}$. Moreover, the above also shows that $p\colon \mathcal{E} \to \mathcal{B}$ is indeed a *locally trivial* smooth principal $\mathrm{Symp}(M, \omega)$-bundle.