Firefox

I did this the hard way by creating my own plugin and then got schooled in the comments on how to actually do it for any bookmark. Link:

https://lemmy.ml/post/1472541?scrollToComments=true

PS: Don't duplicate your post on a bunch of different servers, it just creates a lot of unnecessary spam for people. I can see at least 3 in a row from lemmy.ml.

If you want to be able to add custom engines directly from the settings, in about:config add browser.urlbar.update2.engineAliasRefresh and set the value to true, after that you should see Add button in about:preferences#search

Edit: nvm, there doesn't seem to be field for suggestions...

Edit2: There's also https://mycroftproject.com/, where you can try to find the search engine you want.

That's a good question, after reading the documentation it seems the website that hosts the search has to support the OpenSearch suggestions, then, if it does, you'll probably have to add it when inside the website with the plus icon that appears after clicking on the browser's address bar, last time I checked I think there were a few differences between that and adding it manually by URL template.
I haven't tested this however, so I can't tell you which sites work with it