Who makes the work, the force or the field? This is a bit of a semantic question.
According to the definition of work:

In physics, work is the energy transferred to or from an object via
the application of force along with a displacement. In its simplest form,
it is often represented as the product of force and displacement. A
force is said to do positive work if (when applied) it has a component
in the direction of the displacement of the point of application. A
force does negative work if it has a component opposite to the
direction of the displacement at the point of application of the
force.

That definition seems to imply that the work is done by the force but I usually hear the expression "work done by the electric field on a charged particle".
Strictly speaking, is the work being done by the electric field or by the electric force that the electric field produces on a charge?
 A: The concept of work is based on force and displacement of the material point $P$ of the body the force is acting on:
$$
\Delta W = \mathbf F \cdot \Delta \mathbf s_P.
$$
Origin of the force $\mathbf F$ in body or field is often interesting and valuable, but it is not required to apply the concept of work, say, when using the work-energy theorem.
Sometimes the origin of the force is obvious and then we can change language from "force does the work" to "the origin of the force does the work" because it is often easier to say, it identifies the source of the force, and there is no chance of confusion. For example, as Philip Wood points out, "piston does work on the gas" is easier to say, comprehend and more informative than "piston's force acting on the gas is doing the work". There is no risk of confusion here.
However, in some cases the origin of the force is not obvious or even relevant, and then the concept of work still applies, even if the origin body is not relevant.
For example, pseudo-forces in rotating frames. Centripetal force exists in a rotating frame and it can do work on bodies in that frame when their position in the frame changes according to the standard formula, but there is no obvious origin to this force. It is "just present" due to the choice of the reference frame. So here it would be superfluous to say some body is doing the work, when no such body is present and we just say "centripetal force is doing the work".
A: 
Strictly speaking, is the work being done by the electric field or by the electric force that the electric field produces on a charge?

This is a distinction that makes no difference. If a horse pulls a cart that is attached to the horse via a rope, then it is valid to say “the horse does work on the cart” and it is also valid to say “the tension in the rope does work on the cart”. Both are equally valid statements and neither should cause confusion. Similarly with the field and the force.
A: A force without a field does not exist.
A field exists from itself. A field is real exactly if a material source exists which generates the field.
The force is the strength of the field at the point where it is applied. So it is not the field itself, but a part of the field, expressed as a force, that determines the work it does.

Who makes the work, the force or the field?

The work makes the force as an expression of the field between two objects at the point of application.
