Hi nova ,
It does not seem we can automate a search on formulae.brew.sh
since they don't seem to have any JavaScript running that would "pick up" some kind of query parameter and trigger it.
I took a quick look around and didn't find another site that might be usable.
So, what I will do for now is change the bang to search kagi instead: https://kagi.com/search?q=vim+site%3Aformulae.brew.sh
That seems to produce decent results.
Of course, if anyone else finds a site that we can search directly, let us know!