The estimate you seek is reminiscent of H. Weyl's tube formula. I will give you some pointers referring for more details to section 9.3.5. of these lectures.

Denote by $r$ the distance to $\newcommand{\pa}{\partial}$ $\pa \Omega$ $\newcommand{\bn}{\boldsymbol{n}}$ and by $\bn$ the innner pointing unit normal of $\pa \Omega$. $\newcommand{\bp}{\boldsymbol{p}}$ There exists $r_0>0$ such that the map $\newcommand{\bR}{\mathbb{R}}$
$$
\pa \Omega \times [0,r_0)\ni(\bp,r)\stackrel{\Phi}{\longmapsto} \bp+r\bn\in \bR^n
$$
is a diffeomorphism onto the region $\DeclareMathOperator{\dist}{dist}$
$$A_{\rho_0}:=\big\{\; x\in \bar{\Omega};\;\;\dist(x,\pa \Omega)<\rho_0\;\big\}.$$
Then
$$\int_{A_{\rho_0}} e^{-r^2/t} dx=\int_{\pa\Omega\times [0,r_0)}\Phi^*\big(e^{-\rho^2/t} dx\Big)=\int_{\pa\Omega\times [0,r_0)}e^{-\rho^2/t}\Phi^*(dx).
$$
The pullback of the Euclidean volume form $dx$ via the map $\Phi$ is described explicitly in the above reference. It has the form $\newcommand{\eQ}{\mathscr{Q}}$ $\DeclareMathOperator{\tr}{tr}$
$$
\Phi^*(dx)= \eQ_\bp(r)dV_{\pa \Omega}(\bp) dr,
$$
where, for each $\bp\in \pa \Omega$ $(r)$, $\eQ_\bp$ is a polynomial of degree $n-1$ in $r$
$$
\eQ_\bp(r)=\sum_{j=0}^{n-1}c_j(\bp) r^j=\det\big(1-r S_\bp\big),
$$
where $S_{\bp}$ is the second fundamental form of the hypersurface $\pa \Omega$ at the point $\bp$ *defined in terms of the inner normal* $\bn$. More precisely if $(x^i)$ are local coordinates on $\pa \Omega$ near $\bp$, then
$$
S_\bp(\pa_{x^i},\pa_{x^j})=\big(\; \bn(\bp),\pa^2_{x^ix^j}\bn(\bp)\;\big),
$$
where $(-,-)$ denotes the canonical inner product on $\bR^n$.

Thus
$$\int_{A_{\rho_0}} e^{-r^2/t} dx=\int_0^{r_0} e^{-r^2/t}\left(\int_{\pa\Omega} \eQ_{\bp}(r)dV_{\pa \Omega}(\bp)\right) dr. $$
The integral
$$
K(r):=\int_{\pa\Omega} \eQ_{\bp}(r)dV_{\pa \Omega}(\bp)
$$
appears in Weyl's tube formula and, more precisely $\newcommand{\bom}{\boldsymbol{\omega}}$ (see Eq. (9.3.18) in the above reference)

$$
K(r)=\sum_{k=0}^{n-1} (-1)^{n-1-k}\bom_{n-k}r^{n-k}\mu_k(\Omega),
$$
$$
=\bom_0\mu_{n-1}(\Omega)r-\bom_1\mu_{m-2}(\Omega)r^2+\cdots +(-1)^{n-1}\bom_n\mu_0(\Omega)r^n,
$$
where $\bom_m$ denotes the volume of the $m$-dimensional Euclidean unit ball and $\mu_k(\Omega)$ is the *curvature measure* of degree $k$. (You need to be careful about various sign conventions. In the above reference the second fundamental form is defined using the *outer* normal.)$\DeclareMathOperator{\vol}{vol}$ For example
$$
\mu_{n-1}(\Omega)= \frac{1}{2} \vol_{n-1}(\pa\Omega),
$$
$$
\mu_{n-2}(\Omega)=\frac{1}{2\pi}\int_{\pa \Omega} \tr S_\bp dV_{\pa\Omega}(\bp),
$$
where $\tr S_\bp$ is the mean curvature of $\pa\Omega$ at $\bp$. Also, $\mu_0(\Omega)$ is the Euler characteristic of $\Omega$.

The asymptotics of
$$
J(t)=\int_0^{r_0} e^{-r^2/t}K(r) dr,
$$
can be determined easily by making the change in variables $s=r^2/t$, $r=\sqrt{st}$ so that
$$
J(t)=\frac{\sqrt{t}}{2}\int_0^{r_0^2/t} e^{-s}K(\sqrt{st}) s^{-1/2} ds
$$
$$
=\frac{1}{2}\sum_{k=0}^{n-1}(-1)^{n-1-k}t^{\frac{n-k}{2}}\mu_k(\Omega)\int_0^{r_0^2/t} e^{-s} s^{\frac{n-k}{2}-1} ds.
$$
Observe that as $t\searrow 0$
$$
\int_0^{r_0^2/t} e^{-s} s^{\frac{n-k}{2}-1} ds=\;\underbrace{\int_0^{\infty} e^{-s} s^{\frac{n-k}{2}-1} ds}_{\Gamma\big( \frac{n-k}{2}\big)}\;+ O\big(t^{N}\big),\;\;\forall N>0.
$$
Hence
$$
J(t)=\frac{1}{2}\sum_{k=0}^{n-1}(-1)^{n-1-k}t^{\frac{n-k}{2}}\Gamma\Big(\;\frac{n-k}{2}\;\Big)\mu_k(\Omega) + O\big(t^{N}\big),\;\;\forall N>0.
$$
Finally
$$
|I(t)-J(t)|=\int_{\dist(x,\pa \Omega)>r_0} e^{-\dist(x,\pa\Omega)^2/t} dx
$$
$$
\leq e^{-r_0^2/t}\vol(\Omega)=O(t^{N}),\;\;\forall N>0.
$$
Hence

$$ I(t)=\frac{1}{2}\sum_{k=0}^{n-1}(-1)^{n-1-k}t^{\frac{n-k}{2}}\Gamma\Big(\;\frac{n-k}{2}\;\Big)\mu_k(\Omega) + O\big(t^{N}\big),\;\;\forall N>0.$$

The leading term of this yields the estimate indicated by Carlo Beenakker.

**About the tube formula** The tube formula for smooth domains or convex domains or more generally sets with positive reach are both special cases of the very general kinematic formulas. The Brunn-Minkowski formula is also a special case of the kinematic formula.