$A\subset \mathbb{R}^m$ is an open and bounded set, $f:A\longrightarrow \mathbb{R}^n$, $m\leq n$, is injective, continuously differentiable and its Jacobian matrix has full rank on $A$. Does this suffice for $f$ to be a $C^1$ diffeomorphism onto its image?
Maybe an additional assumption that $f^{-1}$ is continuous would make this enough?
Also is boundedness of $A$ of any help or could it be done away with?
Indeed, these conditions together with the assumption that $f^{-1}$ is continuous are enough, see https://en.wikipedia.org/wiki/Embedding
The first conditions guarantee that $f$ is a local diffeomorphism, but without the latter condition that $f^{-1}$ is continuous, i.e. that $f$ is a homeomorphism onto its image, you can have global problems. The typical example is $A = (0,1)$, $n = 2$ and $f$ is a curve that converges upon itself, for example $$ \lim_{t \to 1} f(t) = f(1/2). $$ Then a neighborhood of the point $f(1/2) \in \mathbb{R}^2$ contains both a neighborhood of $1/2$ and $1$ in $(0,1)$.
Boundedness of $A$ is not relevant for this.