Is there an appraiser trainee or apprenticeship program for military service members and veterans?

Yes. The occupation has two US Department of Labor-approved registered apprenticeship titles: Appraisal and Valuation of Real Property, and Licensed Residential Real Property Appraiser. The Department of the Interior's Appraisal and Valuation Services Office also runs a three-year Appraiser Apprentice Program, open to federal and non-federal staff, that leads toward a Certified General license; it requires a four-year degree, and openings are posted on usajobs.gov.