Trying to answer the question and bring in a few networking basics...
If I read between the lines correctly, you are concerned about performance and generally getting everything to work, but not about security. Let me prefix this by saying that you should probably not worry. If that's all you are concerned with, then forget about the details, pick one single network for all of your devices, and be done with it. Modern switches are good at only putting packets on ports where they are needed, broadcasts don't bother them either, and 20 is not a large number of devices.
From what I understand, (I know very little about networking) I can divide a network by sub-netting [...]
If I statically configured the sensors to have host addresses between 196.128.10 - 196.128.30 and the motorcontrollers to have addresses between 220.127.116.11 - 18.104.22.168 is that considered two different subnets and thus they cannot communicate? Or is this still considered one subnet and they can communicate?
Check out this excellent answer on network masks.
On the MAC level (what we call layer 2, the regime of switches), all of these devices can still communicate; also, there is only one (global) broadcast region on that level.
On the IP level (layer 3, managed by routers), only devices on the same network can talk to each other, and broadcast to each other directly, and you need routing (sic) to talk between networks - which is a can of worms you most likely do not want to open in your specific case.
Or would I want to setup the sensors to broadcast using 22.214.171.124 so that all the motorcontrollers get this data?
This depends on your application and what your devices can do. If your sensors can send to multiple receivers, then you might configure them appropriately. Or you could put a PC inbetween, maybe with some specialized piece of software which "routes" this more at an application level. Or, as you said, use broadcasting.
If I did that and wanted to configure one particular sensor using the PC would I have to temporarily change the sensor IP to a static address? Would the datalogger need to have the broadcast address as to read in all sensor and motorcontroller data/commands?
Yes. A modern switch makes sure to only send packages to the port they need to go. You need to use broadcasting if you want to target multiple recipients. No device "has the broadcast address". It works the other way round: If a packet is sent to the broadcast address, every host on that network will receive it.
Am I even on the rights track? Thoughts? Other suggestions?
My thought is that you are way overthinking this. I would put all of the devices on one convenient subnet and be done with it. If you wish to broadcast, then do so globally if your devices are able to ignore packages not meant for them. Unless you are close to saturating whatever bandwidth you have available with your sensors and controllers, I would not suspect any problems to arise.
Also, be aware that a device (a "physical" MAC) can have multiple IP addresses, in different or the same networks. Simple devices may not be able to do that simply because their software is too simplistic, but any PC can be configured to be part of any amount of networks at the same time. So you could indeed place your devices in individual subnets, have your PC on all of them, and have your PC either do routing on the IP level, or have an application which handles all of this on a higher level.
You can have arbitrarily many subnets (like your 8 /27 subnets) on a single switch without VLANs. VLANs make all of this a bit more complicated and require a configurable switch (more expensive both in the hardware and in time/know-how investment). You would need them if you are in any way considering security. If you are in a trusted space and simply want to get things to work, and have no very special needs which you didn't mention yet, you will not need them.
And at the end of the day, it matters a lot what your devices expect, and are able to do. What has been said about broadcasts above is true on the network level, but any device would need to have the proper software to actually process broadcast packages which arrive at their doorstep in a meaningful way.