They can! Unfortunately, they won't render properly if they're published as a feature layer, but if you publish it as a Map Image Layer, they'll come in just fine.
We don't use annotations, but searching for public anno layers, you can see other orgs who use them in web settings. Shoutout to Chatham County, NC for being the guinea pigs, and for having nice clear annos.
I guess to further address your question, though, how were you planning/hoping to use annotations in a web environment?