WELCOME TO THE FORUM !
your project is rather large and as the old Chinese fortuine cookie said
every journey begins with the first step.
as I see this, you posed a motor and a gear train and a feeder roller.
you need some optics to see the holes in the paper
you need to drive the motor
so, step #1 is to put together a single motor
and the drive mechanisim
and the sprocket
install the optics to see the holes.
I see a DC motor and an encoder.
get that one unit running
I would offer that the placement of the parts would be something you would need to program separately for each part but I would also assume that there would only be a limited number of placement distances. (parts on the tape)
This first one should be a great learning task and we will certainly help you as you stumble and also help guide you to avoid mistakes.
As a comment on the forum, we will help easier when the information is clear and things are not left out. you seem to be the former on that, so that will make all of our lives easier.
since I have a production/manufacturing/field service background, I tend to like a module that is plug and play, so I would tend to have one Arduino on a drive. but, that is not always the case.
it is easier to have one unit fail and 20 others working, and be able to swap one in or out.....
but, much more often, one micro can run all 20 things....
So, as Robin2 said in post#6 start with a single feeder.
break that into modules
tape hole sensor
take a minute and look at every sensor you can imagine you need to run one motor.
list every digital input then every output, and every analog input.
since step one really is to get one line running, your first device needs to be able to handle every I/O
once you have all those listed, add some room for additional lines, then chose between an UNO pin count board and a MEGA pin count board.
Seems like one UNO could handle one motor. An UNO, a NANO and a Mini use the same processor. different packages.
I would suggest you get an official board for your first unit so you have one that you know is made well and proper and it also helps contribute to the operation of this forum.
I also suspect that your first unit will never see your final device and will always be your test unit.
once you get that running, you will know if you will need a single UNO for every motor or if one MEGA will be able to run 20 of them.
as for the X/Y part, there is a lot of tutorials about X/Y tables.
that part is really for a whole different thread.