The question is complicated to answer because the phrase kingdom of God can be used in more than one sense.
In the most general sense, the kingdom of God would seem to be everything under God’s sovereignty, which is all of reality. However, you can’t get away from the kingdom of God in this sense, so most folks mean something more particular when they use the phrase.
In a more restricted sense, one might say that the kingdom of God comprises those things in which God is specially active—i.e., is asserting his sovereignty. In this case, since it requires a miracle every time the Eucharist is celebrated, one would say that the Mass involves the kingdom of God in a special way.
Scripture speaks of the kingdom of God as if it is present in the Person of Christ, and since Christ is present in the Eucharist, the kingdom of God is present in the Mass in that way.
Scripture also speaks of the kingdom of God as if it is an age of history that is inaugurated with Christ. We are living in that age now, the Mass being one of its characteristic features.
Scripture also speaks of the kingdom of God as if it is a future age that begins with the Second Coming of Christ. In this case, since the Eucharist involves a making-present of Christ in anticipation of the Second Coming, it would seem that there is a sense in which the future kingdom of God is made present in the Eucharist.