I asked my social studies teacher to borrow their tape dispenser core (one of the few teachers who even had one in their dispenser). I took it home, and wrote the measurements in my planner. Once I had my dimensions, I started designing on Solidworks. For my first model, i forgot to take into account the hollow parts and rounded edges, so my first attempt was slightly to small and weighted incorrectly. I went back into Solidworks and made the sides wider, and hollowed out the bottom. It fit snug in the tape ring, and spun nicely in the dispenser. Feel free to download the plans (files below) and print it. The dispenser core is very universal, and will fit in almost any standard scotch tape dispenser. I will put the links to some of the tape dispenser it will fit in below.

I used a Prusa i3 mark 2. I had to enable supports to print it. (Each caption for each image is found beneath the image.)     

First I took the dimensions, and sketched them out in solid works. 

Next, I extruded it, and sketched the holder things.

I extruded the holders. 

I printed it, and it was too small. 

I made the whole thing wider first. 

I made the sides wider.

Next I added the side triangle things the squeeze into the tape roll, insuring a secure fit. 

I then made sure all the dimensions matched up with the real tape dispenser core.

I then printed it, and tested it. It ended up working great.