Yes, a single photon can interact with multiple electrons. And it does so in most light-matter interactions.
The simplest example is the scattering of a photon by any atom/molecule with more than one electron. In many introductory textbook on quantum light-matter interaction the absorption of a photon is described to cause a single electron to "jump to a higher shell" or transition into a different orbital. This is a useful approximation for most practical cases, when the exact state of the atom is irrelevant. But a more accurate description of multi-electron systems does not treat the electrons as independent entities, but as part of a non-separable multipartite wavefunction
$$
\Psi \left( \vec{r}_{e_1}, \vec{r}_{e_2}, ... \vec{r}_{e_N} \right) \neq \prod_{i=1}^N \psi \left( \vec{r}_{e_i} \right) \text{.}
$$
So when a photon is absorbed by the multi-electron system it changes the state of the whole system, with all of its electrons.
A different way to look at it is the following: Imagine the photon would change the orbital of only a single electron. Because the probability distribution of the "active" electron changes, the other electrons feel its repulsion in a different shape and therefore adjust their own orbitals. This in turn makes the "active" electron feel a different potential, so it readjusts its orbital, and so on.
Even for systems with well-separated electrons, like a cloud of Hydrogen atoms, collective absorption of a single photon is likely to occur. Compared to the case, where $1$ atom absorbs $1$ photon $|g\rangle |1\rangle \to |e\rangle |0\rangle$, collective absorption of $1$ photon by $N$ atoms happens at a $N$ times faster rate and is therefore the dominating process in an atomic cloud. The resulting state is a coherent superposition of all possible states in which one particular atom absorbed the photon:[1]
$$
|g_1 g_2 ... g_N\rangle |1\rangle \to \frac{1}{\sqrt{N}} \left( e^{i \varphi_1} |e_1 g_2 \cdots g_N\rangle + e^{i \varphi_2} |g_1 e_2 \cdots g_N\rangle + \,\, ... \,\, + e^{i \varphi_N} |g_1 g_2 \cdots e_N\rangle \right) |0\rangle
$$
A more general treatment of this case, including multiple excitations, was first given in R. H. Dicke, Phys. Rev. 93(1) (1953) – Coherence in Spontaneous Radiation Processes.
You can observe photons interacting with many electrons at the same time also in an everyday object – a mirror. This was discussed already in some other questions here,[2 and duplicates] but in a nutshell: If a photon would only interact with $1$ electron of the mirror it would be scattered in all directions. But since the photon extends over many electrons the scattering interferes destructively for most directions. Calculating the direction of positive interference results in the macroscopic law of reflection.
There are also various processes in which a single emitter interacts with multiple photons, but this is another topic.
