It depends entirely on the facility and whoever processes the form on the receiving end.
Some facilities will let something like this slide, particularly if the form is otherwise complete, legitimate, and clearly not an attempt to circumvent the process. A misdirected visitor questionnaire is a minor administrative hiccup, not a security concern, and staff with common sense will sometimes just process it and move on.
Others will reject it on procedural grounds regardless of the circumstances. Facilities that run tight administrative operations tend to require forms be submitted through the correct channel the first time, and no amount of explaining will change that outcome.
There is no way to know which situation you are dealing with until the facility responds. The inmate can submit the form and see what happens. If it goes through, great. If it gets rejected, the only path forward is completing a new application and submitting it the proper way.
The practical advice is to not wait on the outcome before starting a backup. Fill out a new form now and have it ready to submit through the correct channel. If the original gets accepted you have lost nothing. If it gets rejected you are not starting from scratch while visitation approval is delayed even further.