Translation:We do not want to make your mother angry.
You don't have anything that translates from "deine".
I don't know whether "We don't want to upset YOUR mother" would be OK (or is accepted), but I suspect possibly not. AFAIK, sauer when used like this translates to annoyed, angry, pissed off etc, and while it is perfectly valid to use upset to mean angry (although I don't think I ever have myself), the two aren't synonyms; upset can also mean (to make) sad, or most other kinds of mental disturbance (fear, "being shaken up" etc). As such, while it may be fine in context it isn't really the best translation in isolation. Of course that doesn't mean it shouldn't be accepted by Duolingo necessarily.
I answered "we do not want to make your mother angry"and it was accepted, but after reading it again I am not sure if it is right. Literally it seems more like "we want to make your mother not angry?"
These are different because the first implies that your mother is not angry now, and the second implies that she is.
Can someone shed some light?
I translated "we want your mother not to be angry." DL marked as wrong and offered, "We don't want your mother to get angry." my understanding is that "nicht" in this sentence modifies the main verb and not the modal verb - "not be angry" vs "don't want." If this is the case, wouldn't my translation be better than DL's? Or is there a reason why DL's is better?
I'm not qualified to comment on the placement of nicht, but you seem to have the same issue as BlairScots above. Machen means either to make or to do, while you seem to be using it to mean to be (German: sein). "...deine Mutter sauer machen" would mean "...make your mother angry", so regardless of what the nicht applies to your translation is wrong.
Also even assuming that the nicht applies to the sauer, "We want your mother not to be angry" isn't particularly natural English. Normally, rather than saying (to be) not angry*, we'd use an antonym, e.g. "We want your mother to be calm". That said, this being DuoLingo they probably wouldn't accept an antonym. That being the case, it would have to be something like "We want to make your mother not angry."
\If it were used the to be would go in front of the not, although it probably wouldn't be.
I believe that the word order in a German sentence needs to be [subject] [verb] [indirect object] [direct object].
So yes, directly translated it would be "We want your mother not to angry make."
I don't have any German grammar websites handy, but the German dictionary I use is https://dict.leo.org/german-english/. I hope that helps.
(Deutsche Sprache, schwere Sprache. :) )