Oxford English Dictionary 1.a. That takes or supplies the place of another thing or person; substituted instead of the proper thing or person.
2. Of punishment, etc.: Endured or suffered by one person in place of another; accomplished or attained by the substitution of some other person, etc., for the actual offender. Freq. in Theol. with reference to the suffering and death of Christ.
3. Of power, authority, etc.: Exercised by one person, or body of persons, as the representative or deputy of another.
4. a. Performed or achieved by means of another, or by one person, etc., on behalf of another.
d. Experienced imaginatively through another person or agency.
Example from Wiktionary:
People experience vicarious pleasures through watching television.
The definition of vicarious is living as if through someone else or acting for someone else.
An example of vicarious is when a mother who always wanted to be a dancer continually pushes her children to do ballet.