Let's work with virtual bundles. Your question is equivalent to the following:

If we fix a $k \geq 1$, does the map $BU \times BU \rightarrow K(\mathbb{Z},2k)$ representing $c_k \otimes 1 + 1 \otimes c_k$ factor through some map $BU \times BU \rightarrow BU$ composed with the map $BU \rightarrow K(\mathbb{Z},2k)$ representing $c_k$. Recall that $BU$ has a CW structure with cells in every even dimension, and that $c_k$ is represented by a single cell. From this description of $c_k$, the second map $BU \rightarrow K(\mathbb{Z},2k)$ can be constructed by extending the map $BU_{2k}/BU_{2k-1} \rightarrow K(\mathbb{Z},2k)$ ($BU_{2k}$ meaning the 2k-skeleton) which on the summand corresponding to $c_k$ represents a generator of $\pi_{2k}(K(\mathbb{Z},2k))$ and elsewhere is constant, to $BU/BU_{2k-1}$ and then precomposing with the quotient $BU \rightarrow BU/BU_{2k-1}$.

Similarly, $c_k \otimes 1 + 1 \otimes c_k$ can be represented in the same manner where the nontrivial maps in the bouquet of spheres correspond to the cells $c_k \otimes 1$ and $1 \otimes c_k$.

For a CW complex X with cells in only even dimensions and a single 0-cell, there is a one to one correspondence between $[X,BU]_*$ and the direct product generated by by the non-basepoint cells of $X$. This is given by letting the coordinate corresponding to the d-cell, which we name e, take $[f]:X \rightarrow BU$ to the class $S^d \xrightarrow{e} X_d/(X_{d-1}-e) \xrightarrow{f} BU$ (see Atiyah's K-theory Proposition 2.5.2).

In this case we consider $BU \times BU/ (BU \times BU)_{2k-1}$, and we choose an element of the product which in the coordinates corresponding to $c_k \otimes 1$ and $1 \otimes c_k$ is represented by the generator of $\pi_{2k}(BU)$. Let's call the corresponding map $\theta$ from $BU \times BU \rightarrow BU \times BU/ (BU \times BU)_{2k-1} \rightarrow BU$. Up to homotopy we then have a factorization of $c_k \otimes 1 + 1 \otimes c_k$ as $c_k \circ \theta$ because the maps $\pi_{2k}(BU \times BU/ (BU \times BU)_{2k-1}) \rightarrow \pi_{2k}(K(\mathbb{Z},2k))$ are the same. This relies on the (I think true) statement that generator of $\pi_{2k}(BU)$ under the composite $\pi_{2k}(BU) \rightarrow\pi_{2k}(BU /BU_{2k-1}) \rightarrow H_{2k}(BU /BU_{2k-1})$ lands on the cell which represents $c_k$.

$\bf{Edit:}$ Unfortunately, I've read that this is untrue. Instead it lands on some multiple of the dual of the Chern class. Denote this multiple $a_k$.

So to apply the operation to the virtual bundles $E,F$ you take the composite $\theta \circ (E,F)$. As well as having the property $c_k(\theta \circ (E,F))=a_k (c_k (E)+c_k(F))$, we also have that $c_l=0$ for $0<l<k$.