Josh
With the takeup thumbscrew tightened down securely (finger tight, do not use a tool), can you rotate the takeup core on it's shaft? If so, your problem is solved. This is due to two things. First, the plastic cores have an ever so slightly larger inner diameter than the cardboard cores (of which also have 2 varying sizes). If you have new rubber inserts, you generally must use the cardboard cores for takeup. Once the rubber wears just a little bit, you can switch over to the plastic cores and get far more life out of the inserts. Some of the inserts being sent out are actually so "skinny" or short that they won't even grab the cardboard cores. If that is the case, you will have to return them to the dealer you bought them from. If you simply need cardboard cores to use as takeup, I will send you a couple.
<rant mode on>
Just remember, despite the instructions, the takeup cores should NEVER be swapped out until they wear apart, which even if running for 5 shows a day will last a solid year. The trick is when it comes time to change out the media is to simply loosen the takeup thumbscrews, grab the media with your fingers, and pull it right off of the cleaner, leaving the takeup core and all internal parts on the shaft. More parts get lost by those who are in the mindset that you must change out the takeup core between every roll of media. You wouldn't throw away a takeup reel every time you finish running a reel of film onto it, would you? Of course not and the takeup core is no different. 
<rant mode off>
Second, did you make sure you loaded up the media onto the cleaning machine BEFORE you saturated the pads with FilmGuard? Those who dunk the rolls into a cup of FilmGuard (or whatever cleaning solution you are using) and THEN try to secure the head end onto the takeup core are just asking for trouble. The tape just cannot stick with wet media.
If neither one of those solves your problem, then let me know and I'll go over the other possibilities, but it's usually one of these two.
when the rolls are tightened down by hand, they slip and can be rotated by hand quite easily. My question being does this mean I need to get new "grippers?"
Josh
Second, not necessarily do you need new rubber inserts. Try it with a cardboard core first. If it doesn't grip, then worry about getting new inserts.
thanks,
JJ
Load up fresh media onto the cleaner. Make sure the takeup is secured with "magic" tape (masking tape and Artist's tape can slip due to the glue in the tape). Rotate the takeup core at least one full revolution to ensure it "grabs". Once the media is loaded in the fashion above, I grasp the supply roll and try to roll it backward, to verify the takeup will not let go of the media. If it does, re-secure.
Next I set the cleaner upside down (personal preference) on the edge of a table. I have a small trash can underneath it to catch any drips. With one finger from my left hand, I pull the media outward so that the supply roll turns 1/2 of a revolution. Then I gently press the tip of the sprayer against the roll of media and pull the sprayer trigger a few times to saturate the top half of the roll. Next I turn the supply roll back 1/2 turn so that the soaked part of the media is now pointing towards the floor. Then I repeat by soaking the "top" half of it again. Once I feel it is fairly well saturated, I spray another shot or two (if needed) until a drop or two falls off. Then I repeat for the other roll.
Everyone does it a bit differently, but that's how I do it and I never get any FilmGuard on my hands and I can prepare a cleaner in easily under a minute doing it this way.
Paul
Josh
It is not a very good idea to do any modifications in that unit. I can visualize a modification that looks good on paper but could actually make matters worse. I'll explain my case later.
Some think I can get the chain locally. If you send me a picture with a few dimensions, I might be able to find one.
I removed a link and found that it was way too tight...then I tried (and was successful) reforming a few links (look to see if any are deformed) to take up the slack. As I recall, all you need are a pair of needle-nose pliers.
Steve
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"Old projectionists never die, they just changeover!"