Unlike Maaser Rishon, there is no doubt that person receiving Peah is actually eligable to take it. Unlike Terumah and Maaser Sheni, there is no concern about Tahara. So I can't think of a reason that we don't do it today.
The Shulchan Aruch, Yoreh Deah 332:1 writes:
:לקט שכחה ופאה אם אין עניי ישראל מצוים שם ליטלם אין צריך להניחם
In terms of leket, shichacha, and peah if there are no poor members of Klal Yisrael to collect them one need not leave them.
The Rama adds:
(הגה והאידנא אין נוהגין בהם לפי שהרוב עובדי כוכבים ואם יניחום יבואו עובדי כוכבים ויטלום: (טור
Nowadays we don't follow this practice since the majority of people are non-Jews, and if one were to leave out leket, shichacha, or peah the non-jews would take them instead.
According to the Rama we can perhaps understand why peah isn't really observed nowadays outside of Israel.
But we would think it should be still be observed in Jewish communities in Israel since there are many poor Jews around.
halacha.co addresses this, and offers a fascinating suggestion ("לקט שכחה ופאה בזמן הזה") based on the aforementioned Shulchan Aruch, writing that:
Nowadays peah is not left behind because in today's times poor people do not move around from field to field as was done in the past in order to take them.
In other words, if a poor person would come then you would indeed have to give them the gifts. Practically that doesn't happen since poor people don't bother to come to fields anymore.