There's a couple ways to tackle this question, with regards to whether English settlers in the Americas would know of things like lions and elephants.
First, we need to recognize that specifically for an audience of Protestant Christians listening to Edwards, they would be familiar with animals of the Levant named in the Bible (for them, specifically the English-language King James Version). It's a good point here to note that the wildlife of modern day Israel and Palestine is, perhaps unsurprisingly, a mixture of what are considered both African and Asian animals. Even today it hosts populations of gazelles, striped hyenas, oryx antelope, jackals, leopards and 2011 internet's favorite animal, the honey badger. In historic times (and definitely during the period when the texts of the Tanakh/Old Testament were written), this would have included creatures such as Asiatic lions, Syrian elephants (a subspecies of Asian elephants), and cheetahs, which are all now locally extinct.
Lions in particular are a recurring animal in the Hebrew Bible, to the point of them really deserving a whole standalone answer on the history of the image of lions among ancient Hebrews (that gets a bit too far out of my expertise to discuss). Use of lions in verses reaches up towards a hundred separate examples, with it often being used as a simile for human strength. It is also often used as a symbolic animal (as in Isaiah 65 which often gets condensed in its imagery to a lion laying down with a lamb, although the passage in question actually compares a number of predator animals living peaceably with young flock animals). The lion imagery is particularly big in prophetic books such as Isaiah, just mentioned, Jeremiah, and Ezekiel (Ezekiel being met by angelic creatures with four faces: a human's, an ox's, an eagle's and a lion's, and Ezekiel being told about his people "Thou art like a young lion among nations").
However what I strongly suspect to be the Biblical reference Edwards is making to "greedy hungry lions" is no doubt to the Prophet Daniel, and in particular the Book of Daniel Chapter 6, where Daniel is cast by Darius the Mede (who probably didn't exist, by the way) into a den of lions. Daniel's faith in God, and his being cast into the den on false charges, is rewarded:
"My God hath sent his angel, and hath shut the lions' mouths, that they have not hurt me: forasmuch as before him innocency was found in me; and also before thee, O king, have I done no hurt."
I suspect this to be the imagery that Edwards is referencing in particular, because Daniel's faith and holiness literally protected him from hungry lions, where a sinner can expect their fate in Hell to go the opposite.
Anyway, that's the Biblical part of an English-speaking settlers' familiarity with lions. What about a practical familiarity? Eastern North America had plenty of "lions" of a sort, in the form of mountain lions, which are a big cat, but not a member of the Panthera genus to which lions, tigers, jaguars, leopards and snow leopards belong. Currently the only widely-accepted breeding population of mountain lions east of the Mississippi is in Florida (whether there are other breeding populations is actually very controversial), but until the 19th century this cat was fairly widespread. White settlers considered mountain lions, much like wolves, to be not just pests, but evil creatures of the wilderness (using "Wilderness" in the negative Old Biblical sense) to be conquered, vanquished, and eradicated, not merely controlled. Connecticut enacted a bounty on mountain lions in 1684, and Massachusetts in 1742. By the end of the colonial period mountain lions were extirpated from southern New England, but held on in northern New England until the late 19th century. By 1900 they were effectively gone east of the Mississippi, bar in Florida.
As for Panthera leo: despite lions being a native species to the Americas, they were extinct in the New World by 11,000 years or so ago, so white settlers would not have encountered one in the wild. They did have an opportunity to few some in captivity starting in the 18th century though. Captain Arthur Savage is the first person documented to have transported an African lion to North America, which he kept in his home (why not?) for viewing to the public in 1718. He placed an advertisement in the Boston News Letter:
" “All persons having the curiosity of seeing the Noble and Royal beast the Lyon never one before in America, may see him at the house of Capt. Arthur Savage near Mr. Colmans (Brattle Square) church.”
And also placed a sign outside his house reading "The Lion King of Beasts is to be seen here." Apparently some of the lion's fur was even given to the Chief Judge of the Massachusetts Court for a tonic to be made and applied to his sick grandson.
Although zoos would not be established in North America until the first zoo in Philadelphia was opened in 1874, lions would be brought from time to time, usually by merchants or people connected with intercontinental trade, for public viewing from that first time in Boston onwards throughout the 18th century (the first lion in Philly arrived in 1727, apparently). These animals were not just local attractions, but would be taken on tours: a three year old female (Asian) elephant was purchased in Bengal and transported to New York City in 1796. This seems to be the first elephant brought to the United States, but it toured the white-settled parts of the US heavily at that time, hitting every major town and city on the Atlantic seaboard from Newburyport, Massachusetts to Savannah, Georgia.
So: English-speaking settlers in North America would know of such animals as lions and elephants pretty well from a solid reading of the English-language Bible, would have daily exposure to analogous creatures (to lions at least) and as the 18th century wore on they would have increasing opportunities to view individuals of the African and Asian varieties of such beasts. People would actually pay a premium to see in-person animals with Biblical mentions.
Sources:
Conover, Michael R. and Conover, Denise O., "Historical Forces Shaping Americans’ Perceptions of Wildlife and Human-Wildlife Conflicts" (1997). 8 - Eighth Eastern Wildlife Damage Management Conference (1997). 7. https://digitalcommons.unl.edu/ewdcc8/7
Old North Church of Boston here
Davis, Andrew. America's Longest Run: A History of the Walnut Street Theatre
A Biblical Concordance where you can check out references to lions in the King James Version Bible is available here