The goal of my answer is to elaborate on steps (1) and (2) of the accepted answer of Steven Landsburg. In particular, I would like to make clear how a "small enough ideal" looks like and how to produce a non-reducible row like the one in Oblomov's comment. This consists essentially in revisiting the proof of [2, Proposition 1.9], with some K-theoretical shortcuts.
Addendum. I realized that the last paragraph of [2, Proof of Proposition 1.9] is flawed, so that the unimodular row $(21 + 4x, 12, x^2 + 20)$ is not guaranteed to be non-stable in $\mathbb{Z}[X]^3$. Indeed, it is assumed there that $SK_1(\mathcal{O}, 2\mathcal{O}) = \mathbb{Z}/2\mathbb{Z}$ for $\mathcal{O} = \mathbf{Z} + \mathbf{Z} \sqrt{-5}$, whereas $SK_1(\mathcal{O}, 2\mathcal{O})= 1$ by the Bass-Milnor-Serre Theorem (In other words, $f = 2$ does not have the property $(*)$ defined at the top of page 191). This glitch in the final step of the proof is harmless in the sense that changing sligthly the conductor $f$ yields a valid, but different, non-reducible unimodular row. Setting for instance $f = 4$, i.e., $B = \mathbf{Z} + 4 \mathbf{Z} \sqrt{-5} \subset \mathcal{O}$, and considering a suitable quadratic residue symbol, will do the job. My answer below is yet another way to address this problem.
The Bass-Milnor-Serre theorem [3, Theorem 11.33] is key to understand step (1):
let $F$ be a totally imaginary number field and $m$ is the number of elements of finite order in $F^{\ast}$. For each ideal $I$ in the ring of integers $R$ of $F$, the relative special Whitehead group $SK_1(R, I)$ is a cyclic group of finite order $r$. For each prime $p$, $\text{ord}_p(r)$ is the nearest integer in the interval
$\lbrack 0, \text{ord}_p(m) \rbrack$ to
$$\min_{\mathfrak{p}} \left\lfloor \frac{\nu_{\mathfrak{p}}(I)}{\nu_{\mathfrak{p}}(pR)} - \frac{1}{p - 1} \right\rfloor$$
where $\lfloor x \rfloor$
denotes the greatest integer $\le x$ and $\mathfrak{p}$ ranges over the prime ideals of $R$ containing $p$.
Independently of the above theorem, it follows from [1, Corollary 5.2] that $SK_1(R, I)$ surjects onto $SK_1(R, J)$ if $I \subset J$ and $R/I$ is semi-local. For $R$ and $I$ as in the theorem, the group $SK_1(R, I)$ increases in size as $I$ decreases with respect to inclusion. This is reflected in the above formula and it is immediate to see that for every decreasing sequence of ideals $I$, the corresponding sequence $(SK_1(R, I))_I$ stabilizes. Moreover, we can find $I$ such that $SK_1(R, I)$ has the maximal cardinality $m$ and $I$ is then small enough in the sense that any ideal contained in $I$ yields the same maximal $SK_1$ group.
We are now (almost) in position to provide an order $A$ in some $R$ as above such that $\mathbf{Z}[X]$ surjects onto $A$ and $SK_1(A)$ is not trivial. Indeed, consider $R = \mathbf{Z}[i]$, the Gaussian integers, set $A = \mathbb{Z} + 4i \mathbb{Z}$ and $I = 4 \mathbf{Z}[i]$ (this $I$ is sufficiently small for our purpose, but doesn't maximize $SK_1$, unlike $8 \mathbf{Z}[i]$). We will establish that $SK_1(A)$ surjects onto $SK_1(R, I) \simeq \mathbf{Z}/2\mathbf{Z}$, the latter isomorphism being given by Bass-Milnor-Serre's theorem. To do so, let us prove the following lemma:
Let $S$ be a one-dimensional Noetherian domain and let $J$ be an ideal of $S$ such that $S/J \simeq \mathbf{Z}/n\mathbf{Z}$ for some $n \ge 0$. Then $SK_1(S) = SK_1(S, J)$.
Proof.
In the exact sequence [3, Theorem 13.20 and Example 13.22]
$$K_2(S/J) \rightarrow SK_1(S, J) \rightarrow SK_1(S) \rightarrow SK_1(S/J)$$
the last term, namely $SK_1(S/J)$, is trivial since $S/J$ is Artinian and hence of stable range $1$. In addition, the image of $K_2(S/J)$ in $SK_1(S, J)$ is also trivial. Indeed $K_2(S/J)$ is generated by the Steinberg symbol $\{-1 + J, -1 + J\}_{S/J}$ because $S/J\simeq \mathbf{Z}/n \mathbf{Z}$ [3, Exercises 13A.10 and 15C.10]. As $\{-1, -1\}_S$ is a lift of the previous symbol in $K_2(S)$ our claim follows and the proof is now complete.
By the previous lemma we have $SK_1(A) = SK_1(A, I)$. Since the inclusion $A \subset R$ induces an epimorphism from $SK_1(A, I)$ onto $SK_1(R, I)$, we deduce that $SK_1(A) \neq 1$.
It is actually possible to find a non-trivial element in $SK_1(A)$ by considering the power residue symbol $\binom{12}{1 + 4i}_2 = -1$ [2, Section 11C]. This yields an non-trivial Mennicke symbol $[12, 1 + 4i]$ and eventually the non-reducible row $(12, 1 + X, X^2 + 16)$ using the ring epimorphism induced by $X \mapsto 4i$.
Erratum. I claimed erroneously in former versions of this answer that the same technique easily applies to $\mathbb{Z}[X^{\pm 1}]$. My conclusions were wrong and I fail to answer the following
Question.
Is the stable rank of $\mathbb{Z}[X^{\pm 1}]$ equal to $3$?
[1] H. Bass, "K-theory and stable algebra", 1964.
[2] F. Grunewald et al., "On the groups $SL_2(\mathbf{Z}[x])$ and $SL_2(k[x,y])$", 1994.
[3] B. Magurn, "An algebraic introduction to K-theory", 2002.
