Does the Russian Imperial House seek the restoration of the Monarchy in Russia?

No. 

The Grand Duchess repeatedly has said that, as Head of the Imperial House, she would be ready to serve, if the Russian people were to decide to restore the monarchy through a free and open vote organized in a fully legal and constitutional way.  However, she neither actively seeks that position nor believes that Russia at the present moment is ready for a restoration of the monarchy. 

The Grand Duchess believes it is her duty to serve Russia to the fullest extent possible by supporting educational and cultural programs, collaborating with the Russian Orthodox Church and other traditional religious confessions in Russia to improve morals and preserve traditions, and in general to advance in every way open to her the interests of Russia and the Russian people.  The Grand Duchess has repeatedly emphasized that her duty now is to continue to observe and preserve the dynastic laws of the House of Romanoff and to stand ready for the day—should it ever come—to assume the throne that her family has occupied since 1613, but never to support any effort to force herself or the monarchical form of government on the free people of Russia.

As the Grand Duchess put it in an interview in Agence France-Presse (AFP) on September 19, 2007: 

"My main role as Head of the Russian Imperial House is to help the people of Russia to restore the nation’s spiritual and moral values and its national character, to help them build a civil society based on their ancient traditions, which the Revolution nearly destroyed.  In practice, this means our active involvement in charitable, cultural, legal, and other forms of social activities.  At the same time, I am categorically against involving the Imperial House in politics.  The dynasty, whether it rules or not, should be a source of unity, not division.  Ideally, I want each citizen of Russia, regardless of his or her political convictions, social views, or religious affiliation, to know that the Imperial House of Romanov exists and that it belongs to each and every one of them.  That is, I want them to know that there exists a living symbol, which is not just an inanimate representation of the state, like a state coat-of-arms, flag, or national anthem, but is something to which the people can turn for moral support and assistance.  As far as reestablishing the monarchy is concerned, this is not up to me.  Certainly, I believe that the monarchy very much has a future.  For the Imperial House to deny this would be as absurd as the Church denying the existence of God….  Legitimate, hereditary monarchy as a national arbiter that stands above party politics and class interests would be the best guarantor of genuine democracy in the modern world.  That is, in short, my view on the matter.  However, only the people can decide if a monarchy is needed or not.  And the House of Romanoff will always serve Russia no matter what its form of government might be."