Who needs a patriarchate
Apparently the Church since she has had a sort of habit of creating new Patriarchates from time to time.
There is a difference between 'needs' and 'wants' or 'finds useful' or 'allows to happen', all of which would be more accurate characterizations of why and how 'the Church' creates new Patriarchates from time to time.
Ortho_cat has put his finger on why, though in a certain sense it might be technically correct to say that Orthodox are 'sedevacantists', it is actually quite misleading. True sedevacantists are thoroughly Roman, in that they share the same distorted ecclesiology which is at the root of most of the difference between Orthodoxy and the Papacy--to whit, that the "Patriarchate of the West" (that is any honors or privileges possessed by the Bishop of Rome beyond those of any other diocesan bishop) is a integral part of the Apostolic Deposit rather than a historically contingent development with important practical but no doctrinal implications.
The Apostolic portion of Church governance is the bishop ruling his local church, and meeting in council with other bishops to address those issues which affect more than the local church. Everything beyond that is a contingent development that is not necessary to the Faith. Some of those developments are broadly practical: the general organization of those local councils into permanent bodies along geo-political lines and the selection of one see to hold the chairmanship ('presiding') of those bodies. Some simply recognized the contemporary 'facts on the ground' (and then in a conservative organization, those recognitions ossified and remain long after their initial impetus has gone away): Rome, Alexandria and Antioch were the three most important cities in the Roman Empire, with the most people (both Christian and not), and therefore the most resources so they were from early on given greater deference and responsibility; Jerusalem, the city of our Lord's Passion and Ressurection, of the one Church founded by all of the Apostles, of the first martyrs was practically wiped out in 70AD and took a long time to recover, so when it finally did it was slotted in behind the first 3; Constantinople was set up as 'New Rome' so it was given the 'perogatives of honor after Rome'; the Russian Church grew larger than the ancient patriarchates and was backed by an Imperial power, so was made a Patriarchy itself.
So to answer the OP--no, the Patriarchate of the West is not 'vacant'. The institution of the Patriarchate of the West fell into heresy and schism and remains so to this day. Now, when it did so, it left the actual position of Orthodox Bishop of Rome (as well as many other Western cities vacant). The Orthodox did not elect a new Bishop of Rome (or London or Paris or Milan) for largely practical reasons--the Pope had the power to prevent any such bishop from getting anywhere near the actual see, and, more importantly, practically the entirety of the West had followed him into schism so there weren't any Orthodox for such a replacement to shepherd. More recently, as Orthodox have begun to live in Western Europe again (through immigration and conversion), Orthodox bishops have started being appointed to those vacant sees to shepherd them. As ialmisry points out, there is now an Orthodox bishop whosediocese includes the city and faithful of Rome, so there is an Orthodox Bishop of Rome (though that is not his official title, whether because his actual cathedra is elsewhere or in an attempt to be diplomatic or a combination of the two I don't know enough to say). Like all Orthodox bishops, he is a successor of Peter (and Paul and all the Apostles). And in a sense he is a successor to the sainted Orthodox popes of the first millennium as he shares their faith and their communion and cares for the Orthodox in the same region. But he is not the "Patriarch of the West" because the Orthodox Patriarchate of the West ceased to exist a thousand years ago. Should the Orthodox Church in Western Europe ever again grow to the level of being recognized as a Patriarchate, its unlikely the Patriarch would be in Rome--after all, the two major reasons Rome held pre-eminence there in the first millenium (capital of the Empire and an impressive record of fidelity to the Faith in the first 5 or 6 centuries) are long gone.