They can yeah. You can put something like this into the Avatar box in your post flair:
background-repeat: no-repeat;
background-image: url(insert url here);
background-position: 0 0;
background-size: auto 100%;
object-position: 9999px;
There should be some sites (putting "sig rotator" into google should give you options) out there that will let you create a series of images for rotation that you can save as a url. Need to put ?_RNG on the end of the url for it to actual rotate them, or at least that's been how I've had to do it.