Both words would be appropriate to use.
The main difference between 'marriage' and 'wedding' is that the former is the state of being legally joined as husband and wife, while the latter is the marriage celebration.
Although 'marriage' can also mean the official ceremony of being married, the term 'wedding' is normally used for the actual celebration.
My preference would be:
"This is to inform you that Mr. X has made a request for emergency leave from xxxx to xxxx to attend his brother's wedding in (city/state)"
Student or Learner