Evidence that "space exists rather than just particles" What is (theoretical) evidence that you need to define all of space with properties rather than just stating where all particles are? I mean, does every single coordinate x, y, z have a meaning or would it be sufficient to keep track of particle positions and type. In other words, to specify reality, do you need all of
$\Psi(x,y,z,a)$
where $a$ is another dimension (charge, spin, etc.) or is there any framework where you just specify a countable number or particles with coordinates
$(x_{P_i}, y_{P_i}, z_{P_i})$
and you do not care about coordinates inbetween particles (obviously there would be distant interaction laws).
Can you disprove sufficiency of the second representation?
 A: Already to state where your particles are you need space. 
Even if you describe where your particles are by saying how they relate to other particles (e.g., in terms of a primitive notion of distance), this defines coordinates in a space of distances. The set of realizable distances then forms a 3-dimensional submanifold (accirding to the known laws of physics), and you have back your space. (This is roughly how the GPS works.)
Thus only the ''existence'' status might be questionable. But if one questions the existence of the space found coordinatized in the above (or any other manner) then one has reasons to question the existence of everything else in phyics - for the way other physical concepts are identified is always by some elaborate reasoning that gives the concept precision and physical reliability.
(On second reading, the question seems to say - can one disregard the space between the particles? In classicle mechanics of point particles, one can indeed, on a formal level, as the state of the particle system just consists of a list of particle coordinates and velocity coordinates. But a moving particle would then have to create at each moment the point in space it is moving to, a somewhat weird assumption....)
A: A hydrogen atom consists of two particles: a proton and an electron. Can we completely describe the hydrogen atom by saying the proton is here and the electron is there? I don't think even the Bohmians go that far.
A: There are fields and by no means one can say that they are 'contained' in 'points' or 'ensemble of points'.
I think that this is enough as an answer.
Edit add (because of the 1st comment):
Consider this scenario:
There exists only one particle (atom) in the universe. It stays immobile because no external force is applied to it. Is that all? By no means because there exists also the gravitational field that propagates at c speed. How can we measure this speed? The way we do this is using the properties of the atom to measure distances and time. But time is defined with the help of c. And c is a property of the medium (vacuum/space), isn't it so? It must be so, because the light speed is known to be independent of the motions of the emitter and of the receiver then, the c speed has to be relative to the medium. It seems to me that you are asking: How to avoid the medium (as a propagating support) or an equivalent question: How a particle, somewhere away of the source, can know when it is time to react if there is no intervenient space? I will try to answer with an absurd and impossible world:
All particles have to know instantly all events of the universe and keep a record of them. They must have a perfect clock, identical to all other particles, and they must follow in real time all the excitations that will arrive at him (it). Due to the quantum nature it can not make a prevision. In short: any particle must have permanently almost all of the past history of the universe. This kind of scenario calls for a localized infinite amount of knowledge and an instant action at a distance not to say that a coordinate system must be present to any particle and how and where the particle will be moving in the next iteration (a discrete time or space is also a complete mess, IMO ) . 
May be this is a severe argument against your proposal to dismiss the role of 'space'.   
Neither your proposal neither the opposite viewpoint (bellow) are standard, but yours, as shown, is incorrect.
There is Space and this is all.  'Space is the ensemble of the 'Fields' and the Particles and the Light are 'deformations' of the Space that propagate and interact.  
We are used to see mass/charge as confined in particles. 
When we make two gamma rays (light) interact we can get a pair positron/electron that have mass and charge. How to deny that the Light can possess charge? Actually both  charges must be present and a light ray is a 'charge separation' that propagates. It is not a motion of charge, it is only a perturbation of space that is propagating. 
An image you can use (from my friend Alf):
A vibration in a bar of iron propagates and two perspectives can be adopted:
From the perspective of the iron,..., and, more interesting, from the perspective of the vibration.  This vibration is 'free' inside the iron and  happily propagates. If it encounters a hole inside the iron it happens to be an obstacle, the vibration will reflect and/or interfere with itself and continue journey. If if finds the limit of the bar (vacuum) it will reflect in an insurmountable wall.
With this image, where the role of the vacuum and the one of the matter were interchanged, I want to say that the adopted viewpoint is really important. It is not a question of semantics.  
