Under 18 U.S.C. § 922(d), it is unlawful for any person to sell or otherwise dispose of any firearm or ammunition to any person knowing or having reasonable cause to believe that such person has been adjudicated as a mental defective or has been committed to any mental institution.
Possession of a Firearm by the Mentally Ill
The VA letter in the OP was sent to someone in Oregon, so here is Oregon law:
No person shall possess a firearm if he or she:
Was committed to the Oregon Health Authority under ORS 426.130;
Was found to be mentally ill and subject to an order under ORS 426.130 that the person be prohibited from purchasing or possessing a firearm as a result of that mental illness; or
Has been found guilty except for insanity under ORS 161.295 of a felony.
And that prohibits VA from adding the individual to the NICS prohibited persons list how?