Can 1 of 4 children withhold property and bar siblings from their deceased father's apartment on the grounds that she is the only emergency contact listed on lease?

No.  The emergency contact on a lease has nothing to do with who inherits property after someone dies.  However, the landlord probably does not want any part of a dispute between siblings and may allow only the sibling listed on the lease to access the apartment.  As they say, "possession is 9/10ths of the law."  If only one sibling has access, she can do as she pleases.

If you want to challenge this sibling, you will have to hire a lawyer and start probate proceedings.  Otherwise, you have to work it out with her.