Yes it is possible.
By the which database are you using? mdb file means are you using MS Access?
If so build your connection string with password use following
Provider=Microsoft.ACE.OLEDB.12.0;Data Source=DB_PATH;
Jet OLEDB:Database Password=MyDbPassword;
While running application for the first time ask user to provide network path of the mdb file and use it in connection string in place of DB_PATH and connect to database.
Connection string without password
Provider=Microsoft.ACE.OLEDB.12.0;Data SourceDB_PATH;
Persist Security Info=False;