UPDATE I removed the reference to the haftarah as it is not analogous to the megillah. However that discussion is found at Reading the Haftorah From Printed Materials However, I have found references that speak of the other for megillos
Megillas Esther has the requirements of reading from a klaf as part of the mitzvah and one would not fulfill the mitzvah if this were not done. THe other megillos do not have that requirement as part of a mitzvah. When printing became practical, then they began using printed copies just as was done with the Haftara (as seen in the linked article above).
Megilla
11) The entire Megillah must be read from a kosher scroll, written
with proper ink, parchment, markings (sirtut), etc. One who recites
the Megillah by heart has not fulfilled his obligation.
The other megillos are not required to be read from a klaf as pointed out by The Reading of Eicha on Tisha B'Av
This discussion goes into the details of whether or not it requires a bracha. However, during the course of the discussion, we see that the halacha of the other megillos does not require a klaf, unlike megillas Esther. See the Fourth reason of the Rama not to recite a bracha for the other megillos.
The Book of Eicha is one of the five megillot (scrolls). Masechet
Soferim 14:1, states that when one reads one of the five megillot, the
beracha of Al Mikra Megillah is recited. R. David Avudraham, Tefillot
HaPesach codifies the statement of Masechet Soferim. Ramban, Torat
Ha'Adam (Chavel ed. Pg. 258) applies the statement of Masechet Soferim
specifically to the reading of Eicha. Nevertheless, R. Yosef Karo,
Beit Yosef, Orach Chaim 559, notes that common practice is to refrain
from reciting the beracha of Al Mikra Megillah on all of the megillot
with the exception of Megillat Esther.
Rama, Teshuvot HaRama no. 35, addresses the practice of refraining
from the recitation of a beracha. He presents four reasons for this
practice. First, perhaps the beracha of Al Mikra Megillah is only
recited upon obligatory readings of a megillah. If the megillah is
read because of a minhag, one does not recite a beracha. [Rama does
note that the reading of Eicha can be considered an obligatory reading
since it is based on Talmudic sources.] Second, there are different
versions of Masechet Soferim as to what beracha should be recited.
One version is to recite Al Mikra Megillah. Another version is to
recite Al Mikra Ketuvim. Rama suggests that a tradition developed to
omit the beracha in order to avoid this question. Third, Rama
questions whether it is actually appropriate to recite a beracha on
reading a megillah (aside from Megillat Esther). Although Masechet
Soferim does endorse reciting a beracha upon recitation of the
megillot, Rama suggests that the statement in Masechet Soferim is
based on opinions and traditions that are not considered common
practice. Fourth, Rama suggests that one may only recite the beracha
of Al Mikra Megillah if one is reading from a text that is written on
parchment and was written according to the laws of writing a sefer
Torah. Since most communities do not have megillot that are written
on parchment (with the exception of Megillat Esther), they do not
recite the beracha of Al Mikra Megillah.
Rama concludes that one should never recite a beracha on the megillot
(with the exception of Megillat Esther) even if they are written on
parchment. He notes that if the only reason to refrain from reciting
the beracha is that they are not written on parchment, synagogues
would make it a priority to purchase a set of megillot. Since we do
not find such a practice, it must be that one would not recite a
beracha on the megillot even if they are written on parchment.
Magen Avraham 490:9, disagrees with Rama's conclusion and rules that
one should recite a beracha on the reading any of the megillot (except
Kohelet). [Magen Avraham does not seem to require parchment in order
to recite the beracha. Ostensibly, Magen Avraham is following his own
opinion (284:1) that there is no requirement to use parchment for the
Haftorah text (see "Reading the Haftorah from Printed Materials").]
Mishna Berurah 490:19, sides with the opinion of Rama that one should
not recite a beracha on the megillot. However, he notes that one can
justify the practice of reciting a beracha if the megillah is read
from parchment. The Vilna Gaon's personal practice was to read all of
the megillot from parchment and to recite a beracha (see Ma'aseh Rav
no. 175). Those communities that follow all of the minhagim of the
Vilna Gaon recite a beracha on reading the megillah from parchment.
[See R. Yechiel M. Tucatzinski, Sefer Eretz Yisrael 21:2. R.
Tucatzinski implies that reading from a parchment is necessary
regardless of whether one plans on reciting a beracha.]
