Lourdes was built for this. The sick are not an afterthought there; they are carried to the front of every procession, the baths and the grotto are arranged around them, and thousands of volunteers come each year to push wheelchairs. The sanctuary is careful to promise nothing, and so are we; what it offers is a place where suffering is not hidden and prayer for the suffering never stops.
Medjugorje offers confession on a scale found almost nowhere else, with dozens of priests hearing it in the open air each evening. The Divine Mercy Sanctuary in Kraków keeps the Hour of Mercy at three o'clock every afternoon, and St. Faustina's message was written for exactly the people who think they are beyond it. Fátima's children were told to pray for sinners and the suffering; the candlelit Rosary on the esplanade does that still.
If you are coming because of someone, living or dead, you can carry them. A pilgrimage made for another person is an old Catholic practice, and the answers linked below explain how.
