They could presumably draw some kind of nourishment from nature. Although elves did eat, it wasn't necessary to sustain their lives. We are told Elves didn't die from starving, only from being slain and "wasting".
While other answers have been offered along different lines of explanations, I feel this point was glossed over in spite of being mentioned in the question itself
They do not, in fact, need to eat; or they need to eat far less often than other races; or they can derive nourishment from things we can't--spring water? Starlight? Music? (However, the use of Lembas as "waybread" suggests they do need to eat food.)
This I believe is part of the factors that gave Elves their seemingly eternal existence on Earth. Elven hröa (physical bodies) are protected from the natural decay of time that affects other races. So then their Fëar or spirits must be capable of drawing some life-force or nourishment from the environment that offsets the natural decay of biological systems.
This is part of Elven nature - the way they were conceived by <PERSON>, they were literally tied up with the world and much more in tune with it. Throughout the books, we read about the sacredness of nature and natural phenomena to Elves. The light of Eärendil was offered in order to provide spiritual nourishment to <PERSON> in a dark place, so we could imagine that starlight served a similar function for all elven spirits and perhaps even their bodies. 88
In the case of Elves, the two (body and spirit) were much more strongly intertwined.
But we also have in-text references to elven feasting, cooking etc. This could be just cultural since a Silmarillion story has even the Valar feasting and attending parties in Aman. Maybe elves did need to take food to survive, but probably they needed it much less than the mortal races. It's for the same reason that we don't hear of starving as one of the reasons that could lead to physical death for an elf. <PERSON> mentions being slain in war and wasting (due to grief), but not starving.
"It is one with this gift of freedom that the children of Men dwell only a short space in the world alive, and are not bound to it, and depart soon whither the Elves know not. Whereas the Elves remain until the end of days.... For the Elves die not till the world dies, unless they are slain or waste in grief.... But the sons of Men die indeed, and leave the world; wherefore they are called the Guests, or the Strangers." - The Silmarillion, Quenta Silmarillion, Ch 1, Of The Beginning of Days 88