That's an interesting point and almost philosophical again. It boils down to: "What is a proton or electron?" Of course both are first of all concepts to order observations in Nature, and there are different levels of descriptions, all in some way valid at a certain level of accuracy and in some way again invalid if one has a closer look. It's also a question of context, which description is necessary and adequate for describing a certain aspect of natural phenomena.Not exactly, since there are contributions from soft photons and electron-positron pairs, and to a smaller extent also of proton-antiproton pairs (if the proton is taken as elementary)
E.g., a proton (or atomic nucleus) can on the one hand be described as a classical particle if it comes to a useful, however approximate, description of molecules (Born-Oppenheimer approximation) with the electrons binding them together as classical (even static!) electromagnetic fields. It's quite well understood why this approximation works, considering the next precise level of description, namely the molecule with all its "constituents" as non-relativistic quantum particles.
If you look closer at an atomic nucleus you realize it "consists of" protons and neutrons, and you can again ask at different levels of descriptions, how to understand their binding in the nucleus. This reaches (particularly for large nuclei) from a classical fluid-description (Bethe, Weizsächer, Bohr, Wheeler et al) to the nuclear shell model using sophisticated realistic nucleon-nucleon potentials and their derivation from chiral perturbation theory employing renormalization-group methods.
Then you switch again the perspective by going to even higher energies up to deel-inelastic scattering of electrons at a nucleon, and some substructure emerges, described by the "parton model". At even higher energies you resolve also sea quarks and gluons etc. etc.
There's not one answer to "what is a proton" from a theoretical-physics perspective but there's an entire hierarchy of models, each valid within its domain of applicability. What's pretty "stable" across all these levels of description are only a few very fundamental properties like the mass, spin, and various charges.
Even an electron, which is at the level of our knowledge today is still considered an "elementary particle", is not so uniquely described. E.g., an accelerator physicist can treat it usually as a classical point particle or describe it in a continuum-mechanical way (particularly at higher space charges), including some effective way to take into account radiation reaction, which is in principle an unsolved problem in classical electrodynamics. Then the atomic theorist comes far with the idea to use non-relativistic QM or with "relativistic QM" and describe it as a particle with spin 1/2. Then there's of course QFT which is used at higher energies, and particularly if you restrict yourself to situations where QED is sufficient, you can get some way with the perturbative concept of an electron, i.e., a particle with a given mass, spin, and charge which is non-interacting to begin with and then you take into account interactions perturbatively, get into the well-known trouble with divergences and cure them with renormalization (UV) and resummation (IR).
Particularly considering the notorious IR problems, you come to the conclusion that for QED, where you have unscreend and unconfined long-ranged interactions, that the naive perturbative picture and the notion of the corresponding asymptotic free states, is inaccurate, and that one has to use some kinds of other concept. In the picture of the naive perturbation theory you have to "dress the bare electron" with a cloud of soft photons (which we have discussed before in this forum, maybe even in this thread!). This holds even true for the non-realtivistic treatment of "Coulomb scattering" since it's a IR phenomenon.
Concerning the UV problems the answer seems to be Wilson's physical interpretation of the renormalization procedure, i.e., the realization that relativistic QFTs are all effective descriptions at some resolution (or equivalent energy ranges). Whether or not there's a more comprehensive theory from which these effective QFT descriptions can be derived is, as far as I can see, still an open question. At least I don't see a really convincing candidate (given that string theory, as interesting it might be from a mathematical point of view, seems to be completely oversold, because there's no satisfactorial phenomenology derived from it, in contradistinction to the still incomplete concept of QFTs, which are phenomenologically utmost successful).