Lourdes is the shrine best prepared for this. The sanctuary describes its main places of worship as accessible, lends wheelchairs, receives pilgrims who need care at the Accueil Notre-Dame inside the sanctuary when they come with an organized pilgrimage, and lists a service that can provide nursing treatments at a pilgrim's accommodation. These have to be arranged before you arrive.
Carry medications in hand luggage with a written list, check how your insurance treats pre-existing conditions, and build rest into every day. A pilgrimage is prayer and does not replace treatment; do not stop or delay medical care in order to travel. Bold Pilgrim does not provide nursing or medical care.
If travel is not possible now, that is not a spiritual failure: a local shrine, the sacraments in your own parish, or someone making the journey for you are all real ways of bringing your illness to Christ.