This might hardly be a full answer (Sorry), but I'm reading the same textbook too, and I have a strange feeling whenever a "vector field along something" shows up, especially if this "something" guy is so bad as an immersed submanifold. As shown in comments, notations in such cases tend to mislead. I'll try to summarise a few key properties and clarify some pitfalls in Lee's writing, and give a rough idea of a proof (idk if it works).
Set-up
Consider a smooth manifold $M$, with or without boundary. We know that the tangent bundle $\mathrm{T}M$ can be made into a smooth manifold, so that $\pi_M: \mathrm{T}M \to M$ is smooth. Next consider another manifold $S$. By the same arguments we construct the smooth tangent bundle and the smooth projection $\pi_S: \mathrm{T}S \to S$.
Suppose further we have an immersion $j: S \to M$. By definition, $j$ induces injection on every tangent space and is an injection (between sets). However $j$ is not necessarily a topological embedding. That's why I'm very uncomfortable to identify $S$ and its image. With extra data from $j$, we have a new tangent bundle over $S$. We denote $\pi_M^{-1}(j(S))$ by $\mathrm{T}M|_S$. It's a subset of $\mathrm{T}M$. On set-theoretic level, it is safe to write $p_S: \mathrm{T}M|_S \to S$. It is just the composition of $\pi_M|_{TM|_S}$ and $j^{-1}$.
While it is clear $p_S$ is a rough vector bundle over $S$, everything else is not obvious. Two crucial questions appear: (a) Does $\mathrm{T}M|_S$ have a smooth manifold structure such that $p_S$ is a smooth bundle? (b) Is the natural inclusion $\iota: \mathrm{T}M|_S \to \mathrm{T}M$ a smooth immersion?
Being a careless, skimming reader, I've just found question (a) and (b) are taken care of in Lee's Example 10.8 and Example 10.28, which apply to any smooth vector bundle over $M$, including $\mathrm{T}M$. However, even though I believe both properties are correct, Lee's wording confuses me. I'll come back to this in the second part of this answer.
Anyway, two kinds of vector fields of $S$ arise. The first one (often called a vector field on $S$) is a smooth map $V: S \to \mathrm{T}S$, such that $\pi_S\circ V = \mathrm{id}_S$. The second kind (often called a vector field along $S$) is a smooth map $N: S \to \mathrm{T}M|_S$, such that $p_S\circ N = \mathrm{id}_S$. To explain what is a "nowhere tangent" vector field over $S$, we have to make one more assertion: (c) $\mathrm{T}S$ is a smooth subbundle of $\mathrm{T}M|_S$. That is, there exists a smooth, injective bundle homomorphism $i: \mathrm{T}S\to \mathrm{T}M|_S$, such that $p_S\circ i = \pi_S$. [Lee mentioned (c) in Example 10.33. (Proof left as an exercise, as usual :-)] Admitting (c), it's acceptable to identify $\mathrm{T}S$ with its image in $\mathrm{T}M|_S$, and we call vector field $N$ along $S$ "nowhere tangent" if for all $x\in S$, $N(x)\notin \mathrm{T}S$.
Okay, now we can rigoriously state OP's question. Suppose $j: S\to M$ is an immersion with codimension $1$, and $N$ is a nowhere tangent vector field along $S$. If $\omega$ is a orientation form on $M$, then $j^*(N\lrcorner\, \omega)$ is a smooth orientation form on $S$, where $N\lrcorner\, \omega$ is interpreted as a pointwise defined map (smoothness makes no sense here) from $j(S)$ to $\Lambda^{n-1}\mathrm{T}^* M$.
Questions (a) (b) and (c)
(a) We need to construct a smooth structure on $\mathrm{T}M|_S$. Let's look at a toy model: $S = (0,1)$, $M = \mathbb{R}^2$, $j$ is the so-called "figure-8" immersion. For our purpose, we suppose $\lim_{t\to 0} j(t)= \lim_{t\to 1} j(t) = j(0.5) = (0,0)$. Since Lee said the property applies to arbitrary vector bundles, instead of $\mathrm{T}M$, why not consider the trivial line bundle $E \cong \mathbb{R}^3$ over $M$? Our task is to determine $E|_S$ and its smooth structure. [I have a strong feeling that it should be the trivial line bundle over $S$.]
In Example 10.8, what Lee did is described in OP's comment. He stated that $E|_S$ is a (topological) vector bundle as long as $S$ is a subset of $M$. Some chart lemma and that's done. Moreover, as he said, if $S$ is an immersed or embedded submanifold, a smooth structure "follows easily". But I argue that, the immersed submanifold case is subtler. In our example, $S$ should not be viewed as a subset of $M$ at all. If we follow Lee's construction, $E|_S$ will be $j(S)\times \mathbb{R}$. Evidently $j(S)$ is a compact space homeomorphic to figure 8. Thus $j(S)\times \mathbb{R}$ is not even a topological manifold! How could that serve as a smooth vector bundle over $S$?
In my opinion, immersed case should proceed as follows. We assign every point $x$ in $S$ a vector space $D_x = \pi_S^{-1}(j(x))$ and define $D = \bigsqcup_{x\in S} D_x$ with natural projection $p_S: D\to S$. We claim $D$ can be made into a smooth vector bundle over $S$, using the chart lemma (Lemma 10.6).
Recall an immersion is locally an embedding (Prop. 5.22). Therefore, we may find an open covering $U_\alpha$ of $S$, such that for each $\alpha$, the restriction $j|_{U_\alpha}$ is an embedding. (Note that $U$ has a natural smooth manifold structure inherited from $S$.) In addition, we may set $U_\alpha$ small enough, so that $j(U_\alpha)$ is always contained in some trivializable open subset $V_\alpha$ of $M$. By saying $V$ trivializable, I mean there exists a local trivializaition $\Phi: \pi_M^{-1}(V) \to V\times \mathbb{R}^k$. This restricts to a bijection $\pi_M^{-1}(j(U)) \to j(U)\times \mathbb{R}^k$. ($U$ is indexed; I dropped its index $\alpha$ to make the diagram cleaner). Notice that $\pi_M^{-1}(j(U))$ equals to $p^{-1}(U)$. The commutativity of following diagram is immediate: 
This verifies the second condition of the chart lemma. We denote the composition of two horizontal arrows by $\Phi_U$, the local trivialization of $D$ over $U$. As for the third, suppose $U_\alpha, U_\beta \in \mathcal{U}$ have nonempty intersection. By the smoothness of $E$ we have a smooth transition map $V_\alpha\cap V_\beta \to \mathrm{GL}_k(\mathbb{R})$. The transition map we seek is
$$U_\alpha \cap U_\beta\to V_\alpha \cap V_\beta \to \mathrm{GL}_k(\mathbb{R}),$$
which is a pullback by $j$, thus smooth. We now have a smooth vector bundle $D$ over $S$.
(b) There is an canonical injective bundle homomorphism $\iota: D \to E$, induced by inclusions $D_p \hookrightarrow E$. And it's not hard to verify it's an immersion. (The key is to see in local trivialization chart $U\times \mathbb{R}^k \to V\times \mathbb{R}^k$ is a immersion.) But it's definitely not always an embedding, as our figure-8 example suggests. So one must be very cautious about using the notation $D = E|_S$ or the name "inclusion". After all, $E|_S$ and $E|_{j(S)}$ can be very different things.
(c) This is not hard either. Let $U\subset S$ be any open set such that $j|_U$ is an embedding. Then $\mathrm{T}U = \mathrm{T}S|_U$ embeds into $\mathrm{T}M$. It's easy to see $D|_U$ is embedded into $\mathrm{T}M$ by $\iota|_U$. Hence $\iota(D|_U)$ is an embedded submanifold of $\mathrm{T}M$, and (of course) $j_*(\mathrm{T}U)$ is included in $\iota(D|_U)$. The restriction of codomain to an embedded submanifold is smooth (Collary 5.30, also c.f. the remark of Theorem 5.29) Then we have a smooth map from $\mathrm{T}U \to D|_U$, and pushing it out results in a smooth map $\mathrm{T}U \to D$. Finally, it's obvious that different choices of $U$ result the same map on their intersection, so these maps can be glued together into a smooth injective bundle homomorphism $i: \mathrm{T}S \to D$. Or $i: \mathrm{T}S \to \mathrm{T}M|_S$ if you like.
Sketch of a proof
Same trick. Let $U\subset S$ be any open set such that $j|_U$ is an embedding. Because $N$ is a smooth section of $D\to S$, $N|_U$ is a smooth map from $N$ to $D|_U$. Identifying $D|_U$ with $\mathrm{T}M|_{j(U)}$, we obtain a smooth vector field $N_U$ along $j(U)$. Smoothly extend $N|_U$ to a neighborhood $V$ of $j(U)$ (denoted by $\tilde{N}_V$), then do the interior product with $\omega|_V$. Pull $\tilde{N}_V\lrcorner\, \omega_V\in \Lambda^{n-1} \mathrm{T}^*V$ back, using the obivious embedding $U\hookrightarrow V$. Check different choices of $U$ have no effect when they overlap. Glue them up.
Good luck!
Final remarks
I guess my construction in (a) works for weak immersions (having constant rank equal to dimension of $S$, but not necessarily injective), too.
Notation matters. Even more so in differential topology.
Jack Lee and his books are great! Lucid and trustworthy. I don't sense much ambiguity in the smooth manifold book, aside from this vector field subtlety. If he sees this, I really appreciate his comments.
Pandas are cutest. Love from China :-)
