$\newcommand\R{\mathbb R}$Let $$\int_{\R^d}m(x\,1_B,d\mu):=m(x,\mu(B))$$ for all $x\in X$ and Borel $B\subseteq\R^d$. Then, for simple functions $$s_n:=\sum_{j=1}^k x_j\,1_{B_j},$$ with $x_j\in X$ and Borel $B_j\subseteq\R^d$ for all $j$, let $$\int_{\R^d}m\big(s_n,d\mu\big):=\sum_{j=1}^k m(x_j,\mu(B_j)).$$

Then, by approximation, extend this notion of integral to bounded measurable functions $f\colon\R^d\to X$. Here, as in the theory of the Bochner integral, we can use the Pettis theorem implying that a measurable function with values in a separable Banach space is a $\|\mu\|$-almost everywhere limit of a sequence of measurable simple functions, where $\|\mu\|$ is the variation measure for $\mu$ -- cf. e.g. Sections V.4 and V.3 of K. Yosida's Functional Analysis, Sixth Edition.

$\newcommand\R{\mathbb R}$Suppose that $X$ is separable. Let $$\int_{\R^d}m(x\,1_B,d\mu):=m(x,\mu(B))$$ for all $x\in X$ and Borel $B\subseteq\R^d$. Then, for simple functions $$s_n:=\sum_{j=1}^n x_j\,1_{B_j},$$ with $x_j\in X$ and Borel $B_j\subseteq\R^d$ for all $j$, let $$\int_{\R^d}m\big(s_n,d\mu\big):=\sum_{j=1}^k m(x_j,\mu(B_j)).$$

Then, by approximation, extend this notion of integral to bounded measurable functions $f\colon\R^d\to X$. Here, as in the theory of the Bochner integral, we can use the Pettis theorem implying that a measurable function with values in a separable Banach space is a $\|\mu\|$-almost everywhere limit of a sequence of measurable simple functions, where $\|\mu\|$ is the variation measure for $\mu$ -- cf. e.g. Sections V.4 and V.3 of K. Yosida's Functional Analysis, Sixth Edition.

$\newcommand\R{\mathbb R}$Here is a natural idea. Let $$\int_{\R^d}m(x\,1_B,d\mu):=m(x,\mu(B))$$ for all $x\in X$ and Borel $B\subseteq\R^d$. Then, for simple functions $$s_n:=\sum_{j=1}^k x_j\,1_{B_j},$$ let $$\int_{\R^d}m\big(s_n,d\mu\big):=\sum_{j=1}^k m(x_j,\mu(B_j)).$$ Then, by approximation, extend this notion of integral to bounded measurable functions $f\colon\R^d\to X$.

$\newcommand\R{\mathbb R}$Let $$\int_{\R^d}m(x\,1_B,d\mu):=m(x,\mu(B))$$ for all $x\in X$ and Borel $B\subseteq\R^d$. Then, for simple functions $$s_n:=\sum_{j=1}^k x_j\,1_{B_j},$$ with $x_j\in X$ and Borel $B_j\subseteq\R^d$ for all $j$, let $$\int_{\R^d}m\big(s_n,d\mu\big):=\sum_{j=1}^k m(x_j,\mu(B_j)).$$

Then, by approximation, extend this notion of integral to bounded measurable functions $f\colon\R^d\to X$. Here, as in the theory of the Bochner integral, we can use the Pettis theorem implying that a measurable function with values in a separable Banach space is a $\|\mu\|$-almost everywhere limit of a sequence of measurable simple functions, where $\|\mu\|$ is the variation measure for $\mu$ -- cf. e.g. Sections V.4 and V.3 of K. Yosida's Functional Analysis, Sixth Edition.