這兩句的寫法是否都正確？

What is student elections?
It could be elections for student council, for student senate, and etc.

"stand for" has a meaning, in British English, as to try to become elected to a council, parliament, etc. (Longman dictionary)
The sentence should be written as: " I will stand for Student Council."

"stand for" could also mean "to support".
"I will stand for the student elections." means you will support the elections.

"stand in" has a meaning, as to run for the elections 參選
e.g. "I will stand in student elections."

stand in the election
stand for election