Yes, it is true that you can write $f=p\circ\phi$, for a polynomial $p$ and an injective analytic $\phi\colon\mathbb{D}\to\mathbb{C}$. In fact, you can replace $\mathbb{D}$ by any compact subset of $\mathbb{C}$ and the results still holds, where $p$ is instead required just to be a rational function.
First, by expanding as a power series and truncating after $n$ terms, there exists a sequence of polynomials $p_n$ converging uniformly to $f$ on an open set containing $\mathbb{D}$. This is almost enough, applying the following.
Lemma 1: Let $f,\{p_n\}_{n=1,2,\cdots}$ be non-constant analytic functions on an open set $U\subseteq\mathbb{C}$ such that $p_n\to f$ uniformly on $U$. Let $K\subseteq U$ be compact and suppose that, for each $k\ge1$ and $z\in K$ with $f^\prime(z)=\cdots=f^{(k)}(z)=0$ then we also have $p_n(z)=f(z)$ and $p_n^\prime(z)=\cdots=p_n^{(k)}(z)=0$.
Then, for all large enough $n$, there exists ono-to-one analytic $\phi_n\colon K\to U$ with $f=p_n\circ\phi_n$.
This is enough to obtain the required result, so long as $f^\prime$ does not vanish on $\mathbb{D}$, as the condition on the derivatives on $f$ trivially holds. If $f^\prime$ does vanish somewhere on $\mathbb{D}$, then we can still obtain the required result by applying the following consequence of Lemma 1.
Lemma 2: Let $f,\{p_n\}_{n=1,2,\cdots}$ be non-constant analytic functions on an open set $U\subseteq\mathbb{C}$ such that $p_n\to f$ uniformly on $U$. Then, for any compact $K\subseteq U$, for large enough $n$ there exists polynomials $q_n$ and ono-to-one analytic $\phi_n\colon K\to U$ such that $f=(p_n-q_n)\circ\phi_n$.
The polynomials $q_n$ will be of uniformly bounded degree with coefficients tending to zero as $n\to\infty$, although we do not require this here. Once Lemma 2 has been established, our required result follows emmediately by using truncated power series approximations for the polynomials $p_n$. In fact, if $\mathbb{D}$ is replaced by any compact $K$ then the result will still hold where $p$ is instead required to be just be a rational functions. This is because we can uniformly approximate $f$ by a sequence of rational functions $p_n$ on some open set containing $K$, by Runge's theorem. Then, Lemma 2 applies to get the required result.
I'll now give proofs of the Lemmas. Here, I make repeated use of the fact that if analytic functions then their derivates to all orders converge uniformly on compacts (a consequence of Cauchy's integral formula, as explained by David Speyer in his answer).
Proof of Lemma 1:
First, restricting $U$ a small enough open set containing $K$, we can w.l.o.g. suppose that for every $z\in U$ and $k\ge1$ with $f^\prime(z)=\cdots=f^{(k)}(z)=0$ then $p_n(z)=f(z)$ and $p_n^\prime(z)=\cdots=p_n^{(k)}(z)=0$. I'll start by proving the result locally; for each $z_0\in U$ there exists an open neighbourhood $V$ of $z_0$ and a sequence of analytic functions $\phi_n\colon V\to U$ with $\phi_n(z)\to z$ uniformly on $V$ and $f=p_n\circ\phi_n$ on $V$ for all large $n$.
Suppose that $f^\prime(z_0)\not=0$. By rescaling $f$ and $p_n$ if required, we can suppose that $f^\prime(z_0)=1$. For some $r > 0$ the closed ball $\bar B_r(z_0)$ is contained in $U$ and $\Re[f^\prime] > 1/2$ on $\bar B_r(z_0)$. Then, by uniform convergence, $\Re[p_n^\prime] > 1/2$ on $\bar B_r(z_0)$ for large $n$. This implies that $\Re[(p(y)-p(x))/(y-x)] > 1/2$ on the closed ball, so $p_n$ is one-to-one there with derivative $\lVert p_n^\prime\rVert\ge1/2$. This implies that $p_n(B_r(z_0))$ contains $B_{r/2}(p_n(z_0))$, so $p_n(B_r(z_0))\supseteq B_{r/3}(f(z_0))$ for large $n$ and there is a unique (analytic) $p_n^{-1}\colon B_{r/3}(f(z_0))\to B_r(z_0)$ with $p_n\circ p_n^{-1}(z)=z$ (by the inverse function theorem). Choosing an open neighborhood $V$ of $z_0$ such that $f(V)\subseteq B_{r/3}(f(z_0))$, then the analytic functions $\phi_n\colon V\to U$, $\phi_n(z)=p_n^{-1}\circ f(z)$ satisfy the required properties.
Now suppose that $f^\prime(z_0)=0$. Then, there is a $k\ge0$ with $f^\prime(z_0)=\cdots=f^{(k)}(z_0)=0$ and $f^{(k+1)}(z_0)\not=0$. Subtracting a constant from $f$ and $p_n$ if necessary, we can suppose that $f(z_0)=0$. Then, $f(z)=(z-z_0)^kg(z)$ for an analytic function $g$ on $U$ with $g(z_0)\not=0$. By assumption, $p_n(z_0)=\cdots=p_n^{(k)}(z_0)=0$, so we can decompose $p_n(z)=(z-z_0)^kq_n(z)$ where $q_n\to f$ uniformly on $U$. Then, on some neighborhood of $z_0$, $g$ is nonzero and $q_n$ are nonzero for large $n$. Hence, we can take $k$'th roots to obtain analytic functions $\tilde f(z)=(z-z_0)g(z)^{1/k}$ and $\tilde p_n(z)=(z-z_0)q_n(z)^{1/k}$. Furthermore, $\tilde p_n\to\tilde f$ uniformly (so long as we take consistent $k$'th roots for $g$ and $q_n$). We also have $\tilde f^\prime(z_0)=g(z_0)^{1/k}\not=0$. So, by the first case above, there exists an open neighborhood $V$ of $z_0$ and analytic $\phi_n\colon V\to U$ with $\phi_n(z)\to z$ uniformly and $\tilde f=\tilde p_n\circ\phi_n$. Taking $k$'th powers gives $f=p_n\circ\phi_n$.
Now, by compactness of $K$ and the fact that the analytic functions $\phi_n$ exist locally as shown above, there exists a finite cover $\lbrace B_1,\ldots,B_m\rbrace$ of $K$, where $B_i$ are open balls in $U$, and analytic functions $\phi_{in}\colon B_i\to U$ satisfying $f=p_n\circ\phi_{in}$ on $B_i$ and $\phi_{in}(z)\to z$ uniformly as $n\to\infty$.
For whenever $B_i,B_j$ have non-empty intersection then, as $f$ is non-constant, its derivative will be non-zero at some point $z_0\in B_i\cap B_j$ and, wlog, we can suppose that $f^{\prime}(z_0)=1$. Then, by uniform convergence, there is an open neighborhood $B^\prime$ of $z_0$ on which $\Re[p_n^\prime]\ge1/2$ for all large $n$ so, as with the argument above, $p_n$ is one-to-one on this neighborhood. As $p_n\circ\phi_{in}=p_n\circ\phi_{jn}$ this implies that $\phi_{in}=\phi_{jn}$ in a neighborhood of $z_0$ whenever $n$ is large enough that $\phi_{in}(z_0)$ and $\phi_{jn}(z_0)$ are in $B^\prime$. By analytic continuation, this gives $\phi_{in}=\phi_{jn}$ on $B_i\cap B_j$. Therefore, with $V=\cup_iB_i$, we can define analytic $\phi\colon V\to U$ by setting $\phi=\phi_{in}$ on $B_i$. Then, $f=p_n\circ\phi_n$ and $\phi_n(z)\to z$ uniformly.
It only remains to show that $\phi_n$ is one-to-one for large $n$. By uniform convergence of $\phi_n^\prime$ to $1$ on any closed ball $\bar B$ in $V$, we have $\Re[\phi^\prime_n]\ge1/2$ on $\bar B$ for large $n$, so $\phi_n$ is one-to-one on $\bar B$. So, letting $B_1^\prime,\ldots,B_r^\prime$ be open balls covering $K$ and whose closures are in $V$, $\phi_n$ is eventually one-to-one on each $B^\prime_i$. If we let $K^\prime\subseteq\bigcup_iB^\prime_i$ be a compact set with interior containing $K$, then, by compactness, there is an $\epsilon > 0$ such that any $x\not=y\in K^\prime$ with $\lVert x-y\rVert\le\epsilon$ are both contained in some $B^\prime_i$, so $\phi_{in}(x)\not=\phi_{in}(y)$ for large enough $n$. Alternatively, if $n$ islarge enough that $\Vert\phi_n(z)-z\rVert < \epsilon/2$ on $V$ then we have $\lVert\phi_n(x)-\phi_n(y)\rVert\ge\epsilon-\lVert x-y\rVert\gt0$ whenever $\lVert x-y\rVert\gt\epsilon$. So, $\phi_n$ is one-to-one for large $n$.
QED
Proof of Lemma 2:
As explained in David Speyer's answer, Lagrange interpolation gives a sequence of polynomials $q_n$ converging uniformly to $0$ on compacts (as they are of uniformly bounded degree with coefficients tending to zero) such that, for each $z\in K$ and $k\ge1$ with $f^\prime(z)=\cdots=f^{(k)}(z)=0$, then $p_n(z)-q_n(z)=f(z)$ and $p_n^\prime(z)-q_n^\prime(z)=\cdots=p_n^{(k)}(z)-q_n^{(k)}(z)=0$. Then, applying Lemma 1 with $p_n-q_n$ in place of $p_n$ gives the result.
QED
