Deposing an incarcerated person in a civil matter is significantly more complicated than a standard deposition, and most correctional facilities do not accommodate it without a court order compelling their participation.
The standard approach that most facilities will accept is written interrogatories rather than a live deposition. The inmate receives written questions, prepares written answers, and has those answers notarized at the facility. Notarization is available at most correctional institutions through the facility's legal staff or a visiting notary. This process does not require the facility to set up a deposition room or coordinate with outside counsel and is therefore far more likely to be accommodated without resistance.
For an actual deposition, whether in person or by video, you will almost certainly need a court order. Federal courts and most state courts have procedural mechanisms for ordering the deposition of incarcerated witnesses or parties, typically through a writ of habeas corpus ad testificandum if the person needs to physically appear, or through a court order directing the facility to permit a video or telephonic deposition on the premises. Without that order the facility has no obligation to allow outside attorneys access to an inmate for deposition purposes.
On video recording, most jurisdictions permit videotaped depositions in civil matters with proper notice and agreement of the parties, or by court order when consent is not given. An audio-only recording is also permissible in most jurisdictions. The format question is secondary to getting the access question resolved first.
An attorney familiar with civil procedure in the relevant jurisdiction should be handling the motion practice to obtain court authorization before approaching the facility.