Yes, and the approach to getting it approved is worth understanding clearly. This situation came up with my own step-children, and the way to frame the request makes a real difference.
Correctional facilities are evaluated in part by their recidivism rates, meaning how often their released inmates come back. Funding and reputation flow to facilities with low rates, so wardens and administrators have a genuine institutional interest in anything that reduces the likelihood of reoffending. Family connection is one of the strongest protective factors against recidivism that research consistently identifies.
When you contact the facility, reach out to the warden's secretary and frame the request around that. Let them know that these children have known this man as their father their entire lives, that maintaining that bond during his incarceration is important to his rehabilitation and reentry, and that you want to work with the facility to keep that connection intact. Present it as a partnership rather than a demand.
Facilities do make exceptions for meaningful family relationships that fall outside the standard biological or legal definitions when the case is presented thoughtfully and the benefit to the inmate's long-term stability is clear. You do not need legal documentation of step-parenthood to make this argument. You need a compelling, respectful presentation of why the relationship matters and why the facility benefits from supporting it.