A lot of graphics editors and converters support the Google's WEBP nowadays, so it should be quite possible to convert pictures in GIF format to WEBP format. The quality depends on the original source of course and will never improve. Generally speaking though, WEBP file should be smaller than JPEG for example.
Convert Graphics interchange file format to Google image file format.
We found 3 software records in the database related to .gif to .webp file format conversion.
Microsoft Windows software - convert gif to webp on Windows
One of the best bitmap graphic editing program
Convert from gif file format
The tables with software actions are a good pointer for what a certain program does with various file types and often may give users a good hint how to perform certain file conversion, for example the above-mentioned gif to webp. However, it is not perfect and sometimes can show results which are not really usable because of how certain programs work with files and the possible conversion is thus actually not possible at all.
Multiplatform software - convert gif to webp on any platform
Cross-platform batch image transcoder compatible with 500 picture image formats
A popular utility for viewing and converting graphic files