If electromagnetic fields give charge to particles, do photons carry charge?

Remember, electromagnetic field is a distribution of electromagnetic force, not charge. Photon bosons are quantum of this field. So, they are force carriers.. not charge carriers. Only force is exchanged with these messenger particles. Based on this interaction, we determine charge of electrons etc involved. That's it!

The following improvement of your statements eliminates the apparent contradiction:

The electromagentic field is the fundamental entity.

Charges (electrons, positrons, nuclei) are accompanied by (''emit'') an electromagnetic field - a soft virtual photon cloud in terms of QED.

Photons are elementary excitations of the quantum electromagnetic field. They do not carry charge.

A short but technical explanation why photons are chargeless: If the electromagnetic field (i.e. photons) were charged, that would imply non-linear self-interactions such as those occurring for the gluon octet that mediates the strong force. The gluonic Lagrangian takes the form $$\mathcal{L}_\text{SU(3)} = -\frac{1}{2} \, \mathrm{tr}(F^2) = -\frac{1}{4} \, \sum_{a = 1}^8 F_{\mu\nu}^a \, F_a^{\mu\nu},$$ where $$F_{\mu\nu}^a = \partial_\mu A\nu^a - \partial_\nu A\mu^a - g \, f^{a}{}_{bc} \, [A_\mu^b,A_\nu^c].$$ Here, $A_\mu^a$ denotes the gluon fields, $F_{\mu\nu}^a$ their field-strength tensors, $\mu$, $\nu$ are spacetime indices, and $a$, $b$, $c$ color indices. $g$ is the gauge coupling and $f^{abc}$ the structure constants of the $su(3)$ Lie algebra associated with the $SU(3)$ gauge group.

Notice that $F^2$ contains quartic self-interactions among the gluon fields due to the last term. The commutator in the field strength gives the gluons themselves color charge.

And here comes the crucial point: Since $U(1)$ (the gauge group of electromagnetism), unlike $SU(3)$, is abelian, i.e. has trivial group structure with all commutators vanishing, the structure constants of the Lie algebra associated with $U(1)$ are all zero. This suppresses the quartic interactions in the Lagrangian of electromagnetism and prohibits electrically charged photons.

The previous answers are based on classical electromagnetism. If we consider this from a quantum-electrodynamic (QED) standpoint, it's not so simple. In QED, the force between charged fermions is exclusively conveyed by an uncharged boson quanta, photons. This changes the classical problem to "how do the charges know which electromagnetic force to generate, an attracting or repelling one?" I may be wrong about this, but the QED argument I believe is much more direct and leaves the original question open. In fact, I have contributed to this question, because for many years I have been unable to answer it.