Veterans Education and Transfer Extension Act of 2019 or the VET Extension Act of 2019 This bill updates Post-9/11 GI Bill educational assistance for certain veterans or active duty military service members. Specifically, the bill increases the number of months of educational assistance available by the lesser of (1) 15 months, or (2) the amount of months needed to complete remedial and deficiency courses that may be needed prior to beginning a program of education. Under the bill, individuals who do not currently have a dependent, but wish to elect to transfer educational assistance to a dependent, may do so at such a time when they do have a dependent. The bill removes a restriction on the timing of the transfer of educational assistance and also authorizes an individual to designate a new dependent or dependents to receive an unused entitlement.