Yes, mail continues in both directions even during a SHU placement. It is one of the few privileges that survives a trip to the hole largely intact.
Prison administration treats mail as a protected form of communication and makes genuine effort to ensure inmates remain connected to their families and loved ones regardless of disciplinary status. When an inmate goes to the hole, phone access gets severely restricted, commissary gets limited, and movement stops entirely, but the mail keeps moving. Letters, photos, magazines, and books can still be sent and received during segregation.
From the inmate's end, they can still write outgoing letters as long as they have paper, a pen, and a stamp. Indigent inmates are entitled to basic writing materials and postage even in the SHU, so a zero balance does not completely cut off outgoing correspondence.
Incoming mail arrives through the facility's standard distribution process and gets delivered to SHU inmates just as it would in general population, with the same mailroom inspection applying to everything that comes in.
The one exception worth knowing is that courts have in some cases restricted outside contact for inmates convicted of terrorism-related offenses. That is a narrow category that does not apply to the vast majority of people in disciplinary segregation.
If your person is in the hole right now, sending a letter through InmateAid is one of the most meaningful things you can do. Mail is often the only consistent lifeline available during a SHU placement, and having something arrive from home makes a real difference in an environment designed to offer nothing.