To answer the question in the final paragraph: yes, there is such a construction. If $H$ is a closed subgroup of $G$, and if $H$ acts on a $C^*$-algebra $A$, then one defines the induced $C^*$-algebra $\operatorname{Ind}_H^G A$ to be the collection of all continuous, bounded functions $f:G\to A$ satisfying:

1. $f(gh)=h^{-1}f(g)$ for all $h\in H$ and $g\in G$; and
2. the function $gH\mapsto \| f(g)\|$ vanishes at infinity on $G/H$.

$\operatorname{Ind}_H^G A$ is a $C^*$-algebra under pointwise operations and the supremum norm, and it carries an action of $G$ by $*$-automorphisms (coming from the action of $G$ on itself by left translation). This construction really is a generalisation of the situation considered in the earlier part of the question: if $X$ is a locally compact $H$-space then we have $\operatorname{Ind}_H^G C_0(X) \cong C_0(G\times_H X)$, $G$-equivariantly.

Green (Zbl 0407.46053) proved, essentially, that there is a canonical Morita equivalence between the crossed products $(\operatorname{Ind}_H^G A)\rtimes G$ and $A\rtimes H$. An equivalence bimodule can be constructed from a suitable completion of the space of compactly supported continuous functions from $G$ to $A$, similarly to what is done for $A=\mathbb{C}$.

(Incidentally, a small comment on the second-last paragraph of the question: when $X$ is a point I believe that the imprimitivity bimodule is the one that implements unitary induction of representations from $H$ to $G$, as constructed by Rieffel. When $H=G$ this is indeed $C^*(G)$ but I'm not sure this holds in general. For instance, when $H$ is the trivial subgroup the Morita equivalence is between $\mathbb{C}$ and $C_0(G)\rtimes G$, with the equivalence bimodule being $L^2(G)$ (on which $C_0(G)\rtimes G$ acts faithfully as the full $C^*$-algebra of compact operators, per Mackey's generalisation of the Stone-von Neumann theorem).

A good place to learn about all of this, including the history and many related results, is Echterhoff's survey now published as Chapter 2 in Zbl 1390.46001 (also available on the arXiv). See Theorem 2.6.4 in the published version, and Theorem 6.4 in the arXiv version.

To answer the question in the final paragraph: yes, there is such a construction. If $H$ is a closed subgroup of $G$, and if $H$ acts on a $C^*$-algebra $A$, then one defines the induced $C^*$-algebra $\operatorname{Ind}_H^G A$ to be the collection of all continuous, bounded functions $f:G\to A$ satisfying:

1. $f(gh)=h^{-1}f(g)$ for all $h\in H$ and $g\in G$; and
2. the function $gH\mapsto \lVert f(g)\rVert$ vanishes at infinity on $G/H$.

$\operatorname{Ind}_H^G A$ is a $C^*$-algebra under pointwise operations and the supremum norm, and it carries an action of $G$ by $*$-automorphisms (coming from the action of $G$ on itself by left translation). This construction really is a generalisation of the situation considered in the earlier part of the question: if $X$ is a locally compact $H$-space then we have $\operatorname{Ind}_H^G C_0(X) \cong C_0(G\times_H X)$, $G$-equivariantly.

Green (The local structure of twisted covariance algebras, Zbl 0407.46053) proved, essentially, that there is a canonical Morita equivalence between the crossed products $(\operatorname{Ind}_H^G A)\rtimes G$ and $A\rtimes H$. An equivalence bimodule can be constructed from a suitable completion of the space of compactly supported continuous functions from $G$ to $A$, similarly to what is done for $A=\mathbb{C}$.

(Incidentally, a small comment on the second-last paragraph of the question: when $X$ is a point I believe that the imprimitivity bimodule is the one that implements unitary induction of representations from $H$ to $G$, as constructed by Rieffel. When $H=G$ this is indeed $C^*(G)$ but I'm not sure this holds in general. For instance, when $H$ is the trivial subgroup the Morita equivalence is between $\mathbb{C}$ and $C_0(G)\rtimes G$, with the equivalence bimodule being $L^2(G)$ (on which $C_0(G)\rtimes G$ acts faithfully as the full $C^*$-algebra of compact operators, per Mackey's generalisation of the Stone–von Neumann theorem).

A good place to learn about all of this, including the history and many related results, is Echterhoff's survey now published as Chapter 2 in Cuntz, Echterhoff, Li, and Yu - $K$-theory for group $C^*$-algebras and semigroup $C^*$-algebras, Zbl 1390.46001 (also available on the arXiv). See Theorem 2.6.4 in the published version, and Theorem 6.4 in the arXiv version.