Go out for
Become a candidate, apply for something
She WENT OUT FOR a place on the team.
Similar phrasal verbs
Go about Go across Go after Go against Go ahead Go ahead with Go along with Go around Go at Go away Go back Go back on Go before Go below Go by Go down Go down on Go down to Go down with Go for Go for it Go forth Go forward Go in Go in for Go in with Go into Go it Go it alone Go off Go off with Go on Go on about Go on at Go on to Go on with Go one Go out Go out to Go out with Go over Go over to Go past Go round Go through Go through with Go to Go together Go towards Go under Go up Go up to Go with Go without