The Museums' published hours are Monday to Saturday 8:00 a.m. to 8:00 p.m. with final entry at 6:00 p.m., and the last Sunday of each month 9:00 a.m. to 2:00 p.m. (final entry 12:30 p.m.) with free entry. They close on ordinary Sundays and on Vatican holidays listed on the hours page, including January 1 and 6, June 29, August 15, November 1, December 8 and December 25 and 26. The main ticket is the entrance ticket to the Museums and Sistine Chapel; the same site sells the Museums' own guided tours, the Vatican Gardens and early-entry options.
If the date you want shows no availability, four things work. Check the official site again in the days before, since slots are released and returned. Book one of the Museums' own guided tours, which draw on a separate allocation. Go to the Viale Vaticano entrance at opening and buy at the door at the price without online booking, accepting a line. Or book a licensed tour operator's visit with a guide, which costs more than the ticket because you are paying for the guide; that is legitimate, unlike a site that resells the bare ticket at a markup.
Parish and diocesan groups of up to 20 traveling with a priest, deacon or religious can apply for the pilgrimage reduction (10 euros) with a stamped letter from the parish or diocese presented on the day.