You’re unfortunately not going to get far without a ridiculously good format that makes everyone drop their jaws at once.
The last format to get proper adoption were h.264 which took years to get full adoption and even then only when Mozilla found it impossible to hold back. And this was with heavy pushing from browser vendors.
Currently WebM and WebP are the big two formats that are making inroads but even then work is slow and that’s with the push from Google.
If you want any chance to get FLIF as a standard you’re going to need to convince one of the big vendors to back it. You’re going to need a strong case that you’re much better than WebP and I’d say you’d need to focus on getting adoption in the image software space first.
Focus on the image editing software and image viewers, make your format actually usable to the average person. From there push things like the custom codec APIs @AshleyScirra mentioned and the polyfill so people can use FLIF on the web. Once you can show that the format is popular and being used THEN you have a case to take to the vendors.