TTY were only a thing very early on and after that only relevant for cheapniks barely able to afford the computer with no money left to acquire contemporary terminals. That an image from hobbyist computing, not business applicatinsapplications.
Now lets take a look at the titular question:
What did order processing on a teletype look like?
It didn't exist. It was all about optimizing the existing process of a customer filling an order form and some (back) office turning it into a data set.
Just imagine what company would need and thus have a computer system for order handling in like 1970? For sure not a small one doing a dozend orders a day. And any company/department larger then this would already have a punch card based system.
If large enough to have subsidiaries, like distribution centers and/or agents, the next logical step is not to replace the working system by some less capable system with cumbersome teletypes, but improve data entry. A large enough company may already turn the forms a customer fills out into punch card orders as a local site and send a days stack to the central (or next) order processing site, getting back print outs with the stuff ordered whenever the delivery truck comes.
Improvement here is the next logical step, but not by adding extreme costly online communication (for what benefit?), but improve and speed up data entry. This is were systems like the Cogar 4 or Datapoint 2200 came into play around 1970. Local data entry with support for forms, basic referencing and validation, collecting entries and transferring them as batch into the existing mainframe environment. That these system as well could be used as low end data processing was a welcome side effect.
Not to mention that none of this replaces the existing (more or less) central data entry department, like for customers sending in a request letter with a sub from some newspaper add or alike. Similar customers calling in.
While this may of course differ between sectors of business, the basic situation is the same, no matter if auto parts, model railways or corn flakes. For smaller companies, growing into sufficient size, the solution is not siting down and envision a minimalist system, but buying a working solution - as shown.
So, in the end, (almost) no use case for a tty user interface.