It is possible, but it is not guaranteed, and the first step is finding out whether the specific facility allows it at all.
Not every jail or prison permits inmate marriages. Some facilities have a formal process in place and allow ceremonies to take place on site, typically in the visitation area, with a chaplain or approved outside officiant presiding. Others do not allow marriages during incarceration at all, full stop. There is no universal policy across the correctional system, which is why checking with the specific facility first is essential before making any plans.
The right people to contact are the facility chaplain and the inmate's case manager. The chaplain is usually the most knowledgeable about the marriage process, specifically and can tell you whether it is permitted, what the application procedure looks like, and what documentation is required. The case manager oversees the inmate's overall plan and will need to be involved in any formal request that goes to the warden for approval.
On cost, there are two separate expense categories to think about. The first is the civil requirement, which applies regardless of where the ceremony happens. You will need a marriage license from the county where the facility is located, and license fees vary by county but typically run anywhere from twenty to one hundred dollars. The second category is any ceremony costs, which depend on who officiates. A facility chaplain may perform the ceremony at no charge. An outside officiant will have their own fee.
The total cost is modest compared to a traditional wedding, but the approval process takes time, so starting the conversation with the chaplain and case manager well in advance is the smart move.