Newton's third law is always and everywhere true in classical physics as long as it is understood the right way, and that means locally, not action-at-a-distance. The law we are talking about states that forces appear in balanced pairs, the pair we sometimes call "action and reaction". Thus if an electromagnetic force on $A$ causes $A$ to accelerate, for example, then you should expect to find another physical thing, $B$, with an electromagnetic force on it, equal and opposite to the force on $A$. The question now arises, can this $B$ be distant from $A$? Relativity answers an unequivocal "no". The $B$ has to be right next to $A$, because momentum is conserved locally not just globally. So what is this $B$ in the case of an electron being accelerated in an electrostatic field, for example? Answer it is the field itself. When a particle is accelerated by a field, the field itself gains the equal and opposite momentum, and furthermore that momentum appears in each physical location in the field exactly where the particle picked up some momentum. It might seem odd at first to think of a "force acting on a field" but if you prefer you can simply say that the field is acquiring momentum.
When two particles attract one another with equal and opposite forces, the momentum appears locally in the field, such that when it is integrated over all locations there is no net momentum of field (in the chosen frame of reference) and therefore the particles must also be acquiring equal and opposite amounts of momentum. When two interacting particles have forces that are not balanced, as in the example with charged particles whose velocity vectors are at right angles, then the field acquires a net momentum. To repeat, Newton's third law applies perfectly well at each location as long as you apply it directly to the two interacting physical things at each location, i.e. particle and field.
Electromagnetism is the best example to pick for this, because there we can work out an exact relativistic treatment without needing the tools of general relativity. The technical detail is that the 4-divergence of the stress tensor of the field is equal to minus the force per unit volume on the local charged matter: this expresses conservation of energy and momentum together. It is indeed a beautiful aspect of the theory that it captures the notion of momentum conservation so elegantly. It is worth keeping hold of the terminology "law of physics" here (i.e. naming the observation with a name such as "Newton's third law") because it draws attention to a significant and notable fact, namely that force, i.e. that which causes increase of momentum, always appears in balanced pairs. I decided to write quite fully about this in the penultimate chapter of Relativity Made Relativity Easy should you wish to get on top of this at the level of undergraduate but not graduate physics.