"My mom is going to take a vacation next week."

Translation: 我的妈妈下个星期会请假。

[Keith_APP]

Not a good translation.

请假 is to apply for leave from work/school; It is not necessarily equal to taking vacation.

To take leave (without the hint of applying for it) : 放假

To take a vacation (going to countryside, camping, resort, etc., or to another country, for leisure)：去渡假。

[ClayTroyer]

Can't i put the time before the subject in this sentence?

[AbunPang]

下个星期我的妈妈会请假 should be accepted.