GIF, itself, has nothing for accessibility. It’s a (not terribly good; but often good enough) compressed image format.
In the web context, good old ‘alt’ is the go-to " will provide the string specified by ‘alt’ for use by text-based browsers or screenreader software. This has nothing to do with GIF, though, it works exactly the same for any image format the browser supports(and may even work if you specify something invalid as an ‘src’, though I haven’t tried that).
The case of non-blind; but visually sensitive in some way, users would be a trickier one. You would presumably need to know what stimuli are problematic, and filter for those at the GIF renderer layer. I know of no such mechanisms, unless you count the (limited) ‘prevent gifs from looping without user authorization’ features built to keep users from being annoyed, or any manual work done by the creator of the GIF to insure that it is not going to push somebody’s buttons.
Alt, at least on the web, is the way to go(GIF certainly doesn’t, and image formats in general tend not to, provide any internal support for ‘accessibility’ scenarios because they are designed simply to store images, not to cope with all the various ways that somebody might want to not use images or use images with modifications(for photosensitivity, red/green colorblindness, etc.)). HTML, though, is generally supposed to provide for ‘fallback’ or ‘alternative’ display modes for various elements; both because of users with particular needs and because of devices and software with particular limitations.