It is possible to be simultaneously licensed and practicing as a real estate appraiser while also being a licensed and practicing real estate agent. The answer is yes, but there are a couple potential conflicts of interest that you should be wary of.
First of all, there needs to be a clear line between when an individual is acting as an appraiser and when one is acting as an agent. They cannot overlap at all. Obviously, there would be a clear conflict of interest if an appraiser were ever to appraise a home that was also their listing or that they had any type of personal interest in. Appraisers who act as both should be sure to be clear on when they are wearing their agent hat and when they are wearing an appraiser hat.
While there is not always one right answer when it comes to some grey areas that may arise as an appraiser and agent works in both professions, it is important to always disclose any relationships or potential conflicts of interest. How a situation appears to others is also important because even though a professional may not feel something is immoral, a judge may see it differently.