Has any reputable Christian claimed to have been physically visited by Jesus Christ Himself after Paul's conversion?
Ecclesiastical history has many cases of such claims. The real problem here is how to distinguish the seemingly real ones from those that are held to be true according to legend or tradition.
The Catholic Church admits to Jesus visiting people on many occasions, usually in the lives of saints. Even here some are well documented, while other visitations are now reduced to legend and/or tradition.
St. Stephen, the Protomartyr of Christians, like St. Paul had a vision similar to that of St. Paul. However, I am not sure this type of heavenly vision of Jesus is something you would be looking for, so I will simply mention it in passing. Besides,this happened prior to St. Paul's conversion
55 But he, being full of the Holy Ghost, looked up stedfastly into heaven, and saw the glory of God, and Jesus standing on the right hand of God,
56 And said, Behold, I see the heavens opened, and the Son of man standing on the right hand of God. Acts 7:55-56
Still within Apostolic Times, the Catholic Church traditionally that Jesus visited St. Peter while he was fleeing Rome, in order to avoid being martyred. There is a small church on the outskirts of Rome dedicated to this event.
Quō vādis? is a Latin phrase meaning "Where are you marching?". It is also commonly translated as "Where are you going?" or, poetically, "Whither goest thou?".
The phrase originates from the Christian tradition regarding Saint Peter's first words to the risen Christ during their encounter along the Appian Way. According to the apocryphal Acts of Peter (Vercelli Acts XXXV; late 2nd century AD), as Peter flees from crucifixion in Rome at the hands of the government, and along the road outside the city, he meets the risen Jesus. In the Latin translation, Peter asks Jesus, "Quō vādis?" He replies, "Rōmam eō iterum crucifīgī" ("I am going to Rome to be crucified again"). Peter then gains the courage to continue his ministry and returns to the city, where he is martyred by being crucified upside-down. The Church of Domine Quo Vadis in Rome is built where the meeting between Peter and Jesus allegedly took place. The words "quo vadis" as a question also occur at least seven times in the Latin Vulgate.
First documentary evidence for the church comes from the monastery of Sant'Alessio all'Aventino in the 9th century, and describes the church as ubi Dominus apparuit ("where the Lord appeared"). The circumstances of its actual foundation are unknown, but it is reasonable to surmise that it was in response to the legend in the Acts of Peter just quoted. - Domine Quo Vadis
As for other visitations of Jesus to people, according to Catholicism, it seems to be in the hundreds. What this comes down to, is where someone is willing to accept the information at face value or to reject it.
For example, let us take the visions of St. Margaret Mary Alacoque.
St. Margaret Mary Alacoque
In this monastery Alacoque received several private revelations of the Sacred Heart, the first on 27 December 1673 and the final one 18 months later. The visions revealed to her the form of the devotion, the chief features being reception of Holy Communion on the first Friday of each month, Eucharistic adoration during a "Holy hour" on Thursdays, and the celebration of the Feast of the Sacred Heart. She stated that in her vision she was instructed to spend an hour every Thursday night to meditate on Jesus' Agony in the Garden of Gethsemane. The Holy Hour practice later became widespread among Catholics.
On 27 December 1673, the feast of St. John, Margaret Mary said that Jesus had permitted her to rest her head upon his heart, and then disclosed to her the wonders of his love, telling her that he desired to make them known to all mankind and to diffuse the treasures of his goodness, and that he had chosen her for this work.
Initially discouraged in her efforts to follow the instruction she had received in her visions, Alacoque was eventually able to convince her superior, Mother de Saumaise, of the authenticity of her visions. She was unable, however, to convince a group of theologians of the validity of her apparitions, nor was she any more successful with many of the members of her own community, and suffered greatly at their hands. She eventually received the support of Claude de la Colombière, the community's confessor for a time, who declared that the visions were genuine. In 1683, opposition in the community ended when Mother Melin was elected Superior and named Margaret Mary her assistant. She later became Novice Mistress, and saw the monastery observe the Feast of the Sacred Heart privately, beginning in 1686. Two years later, a chapel was built at Paray-le-Monial to honor the Sacred Heart.
In 1689, Alacoque received a private request from Jesus to urge the King of France, Louis XIV, to consecrate the nation to the Sacred Heart, so that he may be "triumphant over all the enemies of Holy Church." Louis XIV, along with his successors, Louis XV & Louis XVI failed to consecrate the nation, and 100 years after Alcoque's vision, Louis XVI was stripped of his power in 1789, during the events of the French Revolution.
Alacoque died on 17 October 1690.
Then there is the vision of Jesus to St. Faustina Kowalska, the Apostle of Divine Mercy.
St. Faustina Kowalska
On February 22, 1931, Our Lord and Savior Jesus Christ appeared to this simple nun, bringing with Him a wonderful message of Mercy for all mankind. Sister Faustina tells us in her diary under this date:
“In the evening, when I was in my cell, I became aware of the Lord Jesus clothed in a white garment. One hand was raised in blessing, the other was touching the garment at the breast. From the opening of the garment at the breast there came forth two large rays, one red and the other pale. In silence I gazed intently at the Lord; my soul was overwhelmed with fear, but also with great joy. After a while Jesus said to me, 'paint an image according to the pattern you see, with the inscription: Jesus, I trust in You.'"
Some time later, Our Lord again spoke to her:
"The pale ray stands for the Water which makes souls righteous; the red ray stands for the Blood which is the life of souls. These two rays issued forth from the depths of My most tender Mercy at that time when My agonizing Heart was opened by a lance on the Cross....Fortunate is the one who will dwell in their shelter, for the just hand of God shall not lay hold of him." - Catholic Apparitions of Jesus to Blessed Faustina, Crakow, Poland: Devotion to the Divine Mercy
The original Image of the Divine Mercy, painted under the guidance of Saint Faustina.
One of the other nuns, Sister Christine, later stated that rays of light from the window were visible that night and attracted the attention of people standing on the other side of the street, implying that it was a "physical" appearance, rather than an interior vision.
I could go on and on giving example from the lives of saints, like of St. Thomas Aquinas himself to St. Anthony of Padua. It should also be be known that there are souls at every epoch in time that Our Lord is visiting somewhere in the world. For Catholics, this is somewhat normal.
Just as a side note, I would like to add the following observation as I find it somewhat interesting.
Some have claimed that Myrna Nazzour, has from December 1982 had apparitions of Our Lady and to a very limited way from Our Lord in the form of allocations. What is interesting here is that she is from Soufanieh, a suburb of Damascus in Syria. This is the very region of St. Paul’s vision of Jesus on his voyage to Damascus.
I know of many situations of ordinary people seeing Our Lord as if in a vision, here and there. One in particular case was in my parish with a nine year old boy dying of cancer, some years ago. The day he died, he said that Jesus was standing at the end of the bed and asked him to join him in heaven. He then asked his mom for permission to die. She took him in her arms and she said yes. He immediately died.