What does the Catholic church say about its own presence or absence when the tribulation starts? Simply the question: What, if anything, does the Catholic Church teach about its existence at the time of the tribulation?
The Catholic Church (CC) believes Christ already reigns on earth through her.  She does not believe in a future rapture at the time of or immediateely preceeding the tribulation and then a 1,000 year millenial reign of Christ.

Some view the idea of tribulation is that it preceeds the millenial reign of Christ for the 1,000 years.  The Catholic Church believes Christ already reigns (through the Catholic Church); there won't be some separate reign when Christ Himself is present on earth. To be clear, she believes in the second coming, but what happens thereafter has nothing to do with the view of a tribulation apart from believers who might be raptured.


  669 As Lord, Christ is also head of the Church, which is his Body. Taken up to heaven and glorified after he had thus fully accomplished his mission, Christ dwells on earth in his Church. The redemption is the source of the authority that Christ, by virtue of the Holy Spirit, exercises over the Church. "The kingdom of Christ [is] already present in mystery", "on earth, the seed and the beginning of the kingdom".
  -CCC-
  
  671 Though already present in his Church, Christ's reign is nevertheless yet to be fulfilled "with power and great glory" by the King's return to earth. This reign is still under attack by the evil powers, even though they have been defeated definitively by Christ's Passover. Until everything is subject to him, "until there be realized new heavens and a new earth in which justice dwells, the pilgrim Church, in her sacraments and institutions, which belong to this present age, carries the mark of this world which will pass, and she herself takes her place among the creatures which groan and travail yet and await the revelation of the sons of God." That is why Christians pray, above all in the Eucharist, to hasten Christ's return by saying to him: Marana tha! "Our Lord, come!"
  -ibid-


So, to answer the OP, the Catholic Church is looking for a "final trial", but one in which she remains present, rather than raptured.  She believes we are already in the millenial reign, but Christ will return.


  675 Before Christ's second coming the Church must pass through a final trial that will shake the faith of many believers. The persecution that accompanies her pilgrimage on earth will unveil the "mystery of iniquity" in the form of a religious deception offering men an apparent solution to their problems at the price of apostasy from the truth. The supreme religious deception is that of the Antichrist, a pseudo-messianism by which man glorifies himself in place of God and of his Messiah come in the flesh.