No. The Grande Chartreuse, mother house of the Carthusian Order in the French Alps, does not admit visitors. The local tourist office states that the place cannot be visited, and the museum's own leaflet explains that the monks' withdrawal means they cannot welcome or meet the public. What a pilgrim can do is visit the Museum of the Grande Chartreuse at Saint-Pierre-de-Chartreuse, housed in the Correrie, the former house of the lay brothers, which presents the monks' life and the Order's history. The museum's leaflet places the monastery two kilometres above the museum and says it can be reached only on foot; the tourist office adds that a walk leads to a viewpoint over the whole complex.
The closure is the point of the place and not a discourtesy. St. Bruno and six companions began the monastery in 1084, when they asked Hugh, the Bishop of Grenoble, for a place of solitude, and the Order takes its name from these mountains. The Carthusians ask on their own site that their solitude be respected, and say that even to close relatives the monks rarely write. Benedict XVI, visiting another Charterhouse at Serra San Bruno in Italy in 2011, told the monks that the Church needs them and that no vocation among God's people is on the fringes.
Plan the visit as a walk and a prayer, not as a tour: go quietly and keep to the paths. The museum's leaflet allows about an hour for the walk. The museum dates from 1957; check its current opening dates and the access arrangements before you travel.
Related questions, answered
Can I make a retreat with the Carthusians?
Not for spiritual renewal alone. The Order states that only prospective candidates to Carthusian life may make a retreat inside one of its monasteries, because the solitary vocation does not allow the monks to receive people who are looking only for spiritual renewal. The same page adds that it is now possible to make a retreat in the spirit of St. Bruno at the Maison Saint Bruno, in the former Charterhouse of Sélignac in France, a house the Order says is managed by the Carthusian Order together with the Diocese of Belley-Ars and which receives men and women in a Carthusian cell. The house's own page asks for a stay of at least three days and for prior experience of silent retreats. It is not a stay with a community of monks, and places and conditions are the house's to decide; ask it directly.
Source: Carthusian Order (chartreux.org): RetreatsChartreuse de Sélignac (selignac.chartreux.org): Carthusian Retreat