"My mom is going to ask for leave next week."
Is the Chinese saying "next week, my mother will ask for leave (to be taken at some unspecified time)"? Would changing the word order to 我的妈妈会请假下个星期。change the sense to "my mother is going to ask for leave, which will be taken next week"? Or is this word order even possible in Chinese? If not, how would the latter be expressed? Is the given phrasing ambiguous, as it is in English?
It is a while since you asked this one, but for future reference if one wanted to emphasise the time that the leave will be taken rather than the time the leave will be requested, you could use 請假 as a separable verb (something that hasn't been taught yet as of this lesson) and put a modifying phrase in between 請 and 假：
我的媽媽會請下個星期的假 = 我的妈妈会请下个星期的假 = My mother will request next week's leave.