A good question, here is a post I wrote recently on this subject:
Regarding John XXIII:
As many posters have already said: many sedevacantists remain unsure about John XXIII, and do not tread there. They will usually just say, "Pope Pius XII was our last certain Pope," leaving open the possibility that John XXIII was a true Pope.
For myself, as it stands now, I believe the evidence supports his claim to the Papacy. As a provisional position, I believe that his claim to the Papacy was most likely legitimate for the following reasons:
1. He was universally and peacefully accepted by the Church.
2. He was accepted by the Roman Clergy whom he ruled.
3. He was peacefully accepted by the hierarchy.
4. He never professed a public heresy.
5. He never taught any heresy or grave error against the Faith.
6. He never promulgated an evil universal disciplinary law.
7. He never signed the documents of Vatican II, he was long deceased before the end of the Council.
8. There is evidence to support the idea that he wanted the Council stopped.
9. The Holy Office under John XXIII continued to function as it should, with Cardinal Ottaviani as it's prefect.
10. All other functions of the Roman Curia and all dioceses continued to function properly and peacefully, during the reign of John XXIII.
11. It is a fact that overall, Catholics throughout the world in the 1950's and up until the death of John XXIII were completely oblivious to the fact that the Church was on the verge of a major revolution. No one grasped this until the ascendency of Paul VI, as the Council was already hijacked and the battle between the orthodox bishops and theologians (periti) vs. the liberals were becoming more public.