Most (perhaps all) states require that you be licensed by the state before you can perform real estate appraisals for a fee, whether the appraisals are for financing, estate valuation, property tax determinations, etc. Google your state's home page and look up which division governs professional licensing for real estate professionals (in Minnesota it's the Department of Commerce, for example). Then check around with local real estate educational providers (for-profit providers and academic institutions) to find out about real estate licensing training.
In all honesty, I think it would be a tad tough to get a part-time gig as an appraiser, and I think you would be well-served to do some informational interviewing with active appraisers in your area to get their insight about the possibilities for part-timers. You might find that the costs (both financial and time) of becoming an appraiser might exceed the benefits if you only want to be part-time.
Also, from what I've seen, real estate appraisers in the commercial area (for home selling and financing) are usually put under a great deal of pressure to hit a certain price on their appraisals in order to get continued business from their contractor (mortgage lending company, real estate agency, etc.), and they are put between a rock and a hard place if they are pressured to overinflate the value of a property: give a realistic value and forget about any further orders from the contractor; OR overinflate the value and risk your appraiser's licence if you were found to have done so deliberately.
JMHO -
A-Jock