= Solution
Over the <Grassmannian> $\operatorname{Gr}_k(\mathbb C^N)$, the <complex tautological bundle on a Grassmannian> is
$$
\gamma_k^N=\{(W,v):W\in\operatorname{Gr}_k(\mathbb C^N),\ v\in W\},\qquad (W,v)\longmapsto W.
$$
Its fibre at $W$ is the k-plane $W$ itself. A neighbourhood of a fixed plane consists of graphs of linear maps $A:W\to W^\perp$; $(A,w)\mapsto(\operatorname{graph}A,w+Aw)$ gives a <local trivialization>. The inclusions of coordinate spaces give the infinite <Grassmannian> and its tautological rank-k bundle $\gamma_k^\infty$.
Use the customary compact Hausdorff base convention for the <classification of complex vector bundles by a Grassmannian>; equivalently the argument uses finite subordinate partitions of unity. Give a rank-k <complex vector bundle> $E\to X$ a continuous <fiber metric> given by Hermitian <inner products>. Choose a finite trivializing cover with local orthonormal coordinates $t_i:E|_{U_i}\to U_i\times\mathbb C^k$, and a subordinate <partition of unity> $\rho_i$ with support contained in $U_i$. Define
$$
J_x(v)=\bigl(\sqrt{\rho_i(x)}\,t_{i,x}(v)\bigr)_i\in\mathbb C^{kr},
$$
extending each component by zero off its support. This is continuous and satisfies $\|J_x(v)\|^2=\sum_i\rho_i(x)\|v\|^2=\|v\|^2$, so it is a fibrewise embedding. The image planes vary continuously: in local frames, their <orthogonal projection matrix> is $J(J^*J)^{-1}J^*$. Thus they define a map $c_E:X\to\operatorname{Gr}_k(\mathbb C^{kr})$, and $v\mapsto(x,J_xv)$ identifies $E$ with $c_E^*\gamma_k^\infty$. This proves that every bundle is a tautological <pullback vector bundle>, by the <finite-dimensional embedding of a complex vector bundle>.
We next prove the two independence assertions that turn this construction into a bijection. First, homotopic maps have isomorphic <pullback vector bundles>. Pulling back along a <homotopy> gives a bundle over compact Hausdorff $X\times I$; embed that bundle in a finite <trivial vector bundle> by the construction just given. Its image has a continuous family of rank-k <orthogonal projection matrices> $P(x,t)$. <Compactness> gives a subdivision $0=t_0<\cdots<t_s=1$ with
$$
\|P(x,t_{j+1})-P(x,t_j)\|<1\qquad\text{for every }x.
$$
Restriction of $P(x,t_{j+1})$ to the image at $t_j$ is injective: if it killed $v$, then $v=(P(x,t_j)-P(x,t_{j+1}))v$, impossible by the strict norm bound unless $v=0$. Equal ranks make it an isomorphism, with continuous local inverses. Composing these bundle maps identifies the endpoint <vector bundles>. The fact that <close orthogonal projections identify their image bundles> therefore makes the pullback assignment well-defined on <homotopy> classes.
Conversely, suppose two classifying maps give isomorphic <vector bundles>. The infinite <Grassmannian> has its standard weak CW topology; the compact-subset property of <CW complexes> puts the compact images of the two maps in finite stages. Their tautological <pullback vector bundles> therefore give two finite-dimensional embeddings $J_0:E\to\mathbb C^{N_0}$ and $J_1:E\to\mathbb C^{N_1}$ of the same bundle, after using the <vector bundle isomorphism>. In orthogonal coordinate blocks interpolate by
$$
J_t(v)=\left(\cos\frac{\pi t}{2}\,J_0(v),\ \sin\frac{\pi t}{2}\,J_1(v)\right).
$$
At least one coefficient is nonzero, so every $J_t$ is injective and its image planes give a continuous <Grassmannian> <homotopy>. One endpoint uses the first block and the other the second block. Moving the latter coordinate copy to the standard one is a fixed unitary change of coordinates, connected to the identity by a path in the <unitary group>. Such a path exists by diagonalizing the <unitary matrix> and continuously varying its <eigenvalue> phases. Hence the original two maps are homotopic in the infinite <Grassmannian>. This also proves independence of all embedding and metric choices. Together the two directions establish
$$
\boxed{\operatorname{Vect}_{\mathbb C}^k(X)\longleftrightarrow[X,\operatorname{Gr}_k(\mathbb C^\infty)],\qquad[f]\longmapsto[f^*\gamma_k^\infty].}
$$
For arbitrary non-Hausdorff compact spaces without the required numerability, that usual bundle-classification statement needs additional hypotheses; the construction above specifies precisely the conventional setting being used.
Write $B=\operatorname{Gr}_k(\mathbb C^\infty)$. For the assumed <fiber bundle> $U_k\to\nu\to B$ with <contractible> total space, the <long exact sequence of homotopy groups of a fibration> has segment
$$
0=\pi_q(\nu)\longrightarrow\pi_q(B)\xrightarrow{\partial}\pi_{q-1}(U_k)\longrightarrow\pi_{q-1}(\nu)=0.
$$
For $q\ge2$ this gives group isomorphisms. For $q=1$, the relevant pointed-set part and connectedness of $U_k$ give $\pi_1(B)=0$, corresponding to the single component $\pi_0(U_k)$. Thus the claimed formula includes its endpoint:
$$
\boxed{\pi_q(B)\cong\pi_{q-1}(U_k)\quad(q\ge1),\qquad\pi_1(B)=0.}
$$
Finally the determinant gives a <fiber bundle> $SU_2\to U_2\to S^1$. Since $\pi_3(S^1)=\pi_4(S^1)=0$, its exact sequence yields $\pi_3(U_2)\cong\pi_3(SU_2)$. The given identification $SU_2\cong S^3$ and the degree calculation $\pi_3(S^3)=\mathbb Z$ therefore give
$$
[S^4,B U_2]=\pi_4(B U_2)\cong\pi_3(U_2)\cong\mathbb Z.
$$
The equality of free and based <homotopy> classes uses the simple connectedness just proved. Hence
$$
\boxed{\operatorname{Vect}_{\mathbb C}^2(S^4)\cong\mathbb Z.}
$$
The classification of <rank-two complex vector bundles over the four-sphere> is a bijection of isomorphism classes, labelled by the integer <clutching function> <homotopy> class after choosing a generator. It is not an assertion that <Whitney sum> gives a group operation on bundles of fixed rank two.
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