Does gravity exist in a vacuum? My understanding has always been that it does from conventional science courses, but really thinking about it, I was wondering if this is really the case. 
To my limited understanding there is a theory that there are gravitons that act as particles to pull two different masses together. If these gravitons really are the physical particles of gravity, then a so called "vacuum" that had gravity wouldn't be a vacuum at all. A real vacuum should lack these particles, and thus, lack gravity? 
Anything in the vacuum should then implode due to its own gravitational attraction within itself? If this is the case, could we say in a real vacuum, external gravity does not exist?
 A: Your intuition is good, but you're mixing up some quantum and classical phenomena.
In classical (i.e. non-quantum) physics, a vacuum is a region of space with no matter. You can have electromagnetic fields in a vacuum, so long as the charges creating the fields are in a different region. By the same token you can have gravitational fields in a vacuum, generated by masses somewhere else in space. In this classical description of the universe, there are no such things as photons or gravitons, and everything (for the most part) works out.
In quantum physics, the story is not so easy. As you say, now our force fields are particles, too (photons and gravitons), so maybe a "quantum vacuum" shouldn't include them either? Unfortunately, it turns out that in quantum mechanics (as rob pointed out) it is impossible to have a perfect vacuum, a state with no particles in it at all. One way to see this is through the energy-time uncertainty principle: $\Delta E \ \Delta t > \hbar/2$.  
A perfect vacuum, a state with no particles at all, must have exactly zero energy. If the energy is exactly zero, then it is completely certain, and $\Delta E = 0$ which violates the uncertainty principle. So the quantum vacuum is not a state with zero particles, it is a state with probably zero particles. And in different situations you may find useful to alter your definition of "probably," so there are a lot of different things physicists will call a "vacuum" in quantum mechanics.
This idea, that quantum mechanically there are always some particles around in any region of space, has some cool consequences that we've verified in the lab! One is the Casimir Effect. This is a force that shows up when you move two objects in a vacuum so close together the pressure from these "virtual" photons causes them to attract. Another is the particle they discovered at the LHC, the Higgs Boson. The Higgs field has a "vacuum expectation value," a perfect quantum vacuum will have a non-zero Higgs field throughout it.  Excitations of this field are the Higgs particles found at the LHC! 
A: You are simply confusing vacuum with "nothingness", which is a philosophical concept.
You can check the definition at wiki

Vacuum is space that is devoid of matter. The word stems from the
  Latin adjective vacuus for "vacant" or "void". An approximation to
  such vacuum is a region with a gaseous pressure much less than
  atmospheric pressure.[1] Physicists often discuss ideal test results
  that would occur in a perfect vacuum, which they sometimes simply call
  "vacuum" or free space, and use the term partial vacuum to refer to an
  actual imperfect vacuum as one might have in a laboratory or in space.

There are different theories that try to explain gravitey (curvature of space-time, graviton, etc) but according none of this gravity or gravitons can be considered matter
A: The graviton is the hypothetical gauge boson associated with the gravitational field. I say hypothetical because it is far from clear whether gravity can be described by a quantum field theory, so it isn't clear whether gravitons are a useful description.
In any case, you should not take the notion of virtual particles like the graviton too seriously. have a look at Matt Strassler's article on virtual particles. Virtual partices are really just a mathematical device for describing the energy in quantum fields. So even if the graviton is a good description of gravity we shouldn't view the vacuum as being full of gravitons and therefore not really a vacuum.
For example, suppose we put a charged particle in a vacuum. Would you claim the vacuum is not a vacuum because there is an electrical field in it? If so then you would also have to say the vacuum near a massive body isn't a vacuum because there is a gravitational field in it. While I suppose there is some validity to this claim, it seems excessively zealous.
A: In quantum mechanics, it's impossible to remove all the particles from a vacuum. A volume of space time that contains only photons and gravitons in thermal equilribium (or not) sounds like a perfectly good vacuum to me.
A: A perfect vacuum never exists as mentioned in multiple other comments. All "messenger particles" are fluctuations of their respective fields (e.g. the graviton a place in the gravitational field that has a non zero energy value). All fields are subject to quantum fluctuations, in essence, they rarely have no energy at one point but the fluctuations average to be zero (that is for most fields, others such as the proposed Higgs field possibly have non-neglibile energy values at their lowest energy state).
Since the graviton can also be described as a wave function (much like light; there is theoretically such a thing as gravity waves that warp space-time). 
This and the point made earlier are some proof why there is no such thing as a perfect vacuum.
What may make the situation a bit more complicated is string theory which predicts the graviton to be a close ended string suggesting its ability to interact with more than our three spacial and one time dimension.
(All information summarised from Brian Greene's Fabric of the Cosmos
A: I believe that part of the problem is not having a clear definition of "vacuum."
I can think of at least three types of vacuum.  1) absolute 2) conventional & 3) "practical" vacuum. The practical vacuum is the type you find in a "lab."  The conventional vacuum is the one defined as the "absence of matter."  The absolute vacuum does not exist, other than "theoretically."
Using the practical and conventional definitions for vacuum, the answer to the question is yes, gravity exists in these types of vacuum.  For the absolute definition, the answer is no, because nothing exist (not even fields, photons, fluctuations, gravitons, etc.). 
A: Yes, gravity does exist in a vacuum. A vacuum does not need to be completely devoid of matter, it just needs to have a lower pressure than the area around it.

Consider the syringe above. If I was to put my finger over the end, and then pull on the plunger, an imperfect vacuum would be created. If there was a solid mass in the syringe cavity, it would still obey gravity.  
