Let $M$ be a smooth manifold of dimension $m$ and $\phi : S \to M$ a smooth embedding (dim S = k < m) such that the normal bundle $T_S M$ is trivializable. Let $f: T_S M \to S \times \mathbb{R}^{m-k}$ be a framing of $T_S M$. According to what I'm reading, this should induce an "isotopy class" of embeddings $\Phi : S \times \mathbb{D}^{m-k} \to M $ so that $\Phi( S \times {0}) = \phi(S)$.
This is my first encounter with the word "isotopy". From Wikipedia, if $f, g : X \to Y$ are embeddings, an isotopy from $f$ to $g$ is a homotopy $F: X \times [0,1] \to Y$ such that $F( *, t)$ is an embedding, for each t. I'm unclear if what's meant in the first paragraph is that the map $\Phi$ is only unique up to isotopy, or, if more broadly, we're extending the definition of isotopy here to mean that $\Phi( *, z)$ is an embedding, for every $z \in \mathbb{D}^{m-k}$.
My second question involves justifying the statement made in the first paragraph. What I've worked out so far is the following: by the tubular neighborhood theorem, after identifying $T_S M$ with $S \times \mathbb{R}^{m-k}$ via $f$, I can find an open set $U \subset S \times \mathbb{R}^{m-k}$ containing $S \times \{0\}$ and an open set $V \subset M$ containing $S$ and a diffeomorphism $ \psi: U \to V$ so that $\psi(s,0) = s$, for all $s \in S$. If $S$ is compact, I could then find an open ball $B \subset \mathbb{R}^{m-k}$ containing $0$ so that $S \times B \subset U$ and then I would just restrict $\psi|_{S \times B}$ to obtain the desired embedding.
If S is not compact, I'm not so sure I can do this. Does the statement still hold when $S$ is not compact?
Isotopy just means you have a homotopy trough embeddings, as you say. Concerning the other question, your argument in the compact case works in general, using "balls of variable radii". Consider an open neigborhood $W\subset X=S\times\mathbb R^{m-k}$ of $S\times\{0\}$. Then the function $$ \eta:S\to\mathbb R:x\mapsto \eta(x)=dist((x,0),X\setminus W) $$ is continuous and never zero on $S$. By approximation, there is a smooth function $\varepsilon:M\to\mathbb R$ with $0<\varepsilon<\eta$, and $$ W_\varepsilon=\{(x,y)\in X:dist((x,0),(x,y))^2=\|y\|^2<\varepsilon(x)^2\}\subset W $$ is an open nbhd of $S\times\{0\}$. This nbhd is diffeo to $S\times\mathbb R^{m-k}$ via $$ (x,y)\mapsto\big(x,\tfrac{y}{\sqrt{\varepsilon(x)^2-\|y\|^2}}\big) $$