We have two Polygons/Polyhedra P and Q.
Does there exist a polynomial time algorithm to decide if we can put P (using translation and rotation) inside Q or not?
First think about the case of which P and Q are convex and the input is the list of vertices, edges and faces of P,Q.
To make sense of this question one has to say first what you mean by "copy of $P$". For example one could image to allow translations, rigid motions and/or dilations.
The answer also depends on how the the (convex) polytopes are given.
There is some literature on this topic, a starting point could be:
Peter Gritzmann and Victor Klee. On the complexity of some basic problems in computational convexity: I. containment problems. Discrete Mathematics, 136(1):129–174, 1994.
From this paper:
This Theorem does not consider a "copy of $P$" but only $P$ itself, other cases lead to cases that don't have a polynomial time algorithm.
For the polygon-case a good reference might be:
Bernard Chazelle. The polygon containment problem. In Franco P. Preparata, editor, Advances in Computing Research I, pages 1–33. JAI Press, 1983.
Edit: after your clarification let my restate the question:
Given two $\mathcal{V}$-polytopes $P$ and $Q$, is there a polynomial time algorithm to find a $\mathcal{V}$-polytope $P'$ such that $P'\subset Q$ and $P'$ can be transformed into $P$ by rigid motions?
The answer to this question is: No
The relevant section of the first reference above is Section 7. Optimal containment under similarity and related questions.
As mentioned there, this question is already difficult if $Q$ is a cube. Note the they look at the closely related question of finding a copy of $Q$ that contains $P$. (Instead of "contains $P$" we can simply ask for "covers the vertices of $P$", if convexity is assumed)
There is quite a bit of literature specifically on polygon containment, much of deriving from robotics applications. The paper below solves the containment problem, allowing translations and rotations, with a polynomial-time algorithm (roughly $O(n^6)$).
Avnaim, Francis, and Jean Daniel Boissonnat. "Polygon placement under translation and rotation." Informatique theorique et applications. 23(1). 1989. p.5-28.
Here is a snapshot from their introduction. The two polygons, named $E$ and $I$, have $n$ and $m$ edges respectively. This Intro reviews related, prior work:
