Why can protons collide? Electromagnetic repulsion is inversely related to the square of the distance between both objects.  Shouldn't this mean that the integral of the force between two protons is infinite if the distance becomes zero?  If so, how can protons fuse?  Or does proton fusing occur while there is space between the protons?
 A: 
Shouldn't this mean that the integral of the force between two protons is infinite if the distance becomes zero? 

Physics developed quantum mechanics to  model nature, as it became obvious that the infinities described by classical 1/r potentials would never build up atoms and build up the world we know. The attraction of the electrostatic potential would absorb the electrons into the nucleus and a hydrogen atom would never exist, nor chemical bonds.
As spectroscopic data from experiments showed the electron of the hydrogen atom  , instead of spiraling down on the proton  radiating continuously, generated a spectral series which was fitted with the Balmer etc series. The Bohr model was born which led to the complete theory of Quantum mechanics.
In quantum mechanics, a proton impinging on a proton feels a repulsion and scatters off, the crossection of the scattering can be accurately calculated as a function of the energy of the projectile. The experiment showed that when the energy became large enough the electrostatic potential no longer repelled all the projectiles, but some of the impact  "enters"  the proton itself and scatters off hard cores within the proton. The original experiments probing the constituents of the proton were with electrons, but the theory is the same.  That is how quarks were discovered. 
Edit after comment by OP
Deep inelastic scattering is represented in Feynman diagrams as:

From left to right the incoming projectile interacts   by exchanging a transfer of energy and momentum  represented by the wiggly line as a virtual particle   to the target proton, represented as a ball, and it interacts directly with a constituent. (The projectile can be another ball as the icon for the target proton)  For this diagram the energy is high enough so that the individual constituents of the target  proton  have a probability to interact directly, instead of  as a whole, and their behavior can be studied. 
In proton proton scattering at low energies the exchanged virtual particle is electromagnetic, and the scattering crossection will show the repulsive force between two same charges. As the energy grows, the strong force between the constituents takes over, and virtual gluons are exchanged, overcoming any effect from the virtual photons which reflect the repulsion of like charges.

Or does proton fusing occur while there is space between the protons?

In proton proton scattering  as a function of the energy, at center of mass energies close to the binding energy of  the  Deuteron, below two MeV,  the weak interaction also plays a role in addition to the electromagnetic: there is a probability of fusion into a deuteron with a positron  and an electron neutrino taking up the energy, momentum and quantum number balance. Due to the weakness of the weak interaction this is practically not measurable in the lab and mainly theoretical calculations are used. This process is important for astrophysical calculations of star evolution, where densities and temperatures are high enough to generate  the reaction.
