This can be achieved with a touch of Google-fu.
We want to limit our Google search to search only the site,
From reading the url structure of each result, we notice that definitions all contain
?term=, so can we refine the search with these bits of info:
Then, we add a space and the term we are looking for; if it appears in the text describing a word's etymology, we have a hit.
For example, we'd type the following if we wanted to search for phagos:
site:etymonline.com inurl:term phagos
Search results for "phagos"
We are a touch limited in that we must rely on the definitions containing that particular variant. For example, the above search returns 5 hits; however, a search for phagous returns 13 hits despite phagos and phagous sharing a common root.
Hope that helps!
EDIT: I've further played with this and noticed that occasionally it returns search pages which don't really add much.
These can be filtered out as they all contain the expression
?search=, so we can use:
site:etymonline.com inurl:term -inurl:search phagos
For anyone interested in understanding how that works, prepending a
- negates the statement so
-inurl:search evaluates to AND url does not contain "search".