Currently I'm not using a translation portal, the JSON files are edited by hand: https://wiki.openstreetmap.org/wiki/OpenStreetBrowser/Howto_translate
Weblate looks really nice, I was looking for something like this. I will look into installing it for OpenStreetBrowser. I hope I can set it up in a few days, please stay tuned :-)
It's done! I just finished setting up Weblate. I really like it :-)
One module can't be translated via weblate, as it needs actual source code: https://github.com/plepe/openstreetmap-date-format
Have fun! If have have any comments, I'd like to hear them ...