Great question. Jonathan Edwards would have been a member of his local Congregational ministerial association, and while local associations of churches were in their infancy in the first third of the 1700s, they became normal in Massachusetts by the late 1700s and were normal in Connecticut at an earlier date.
I'd normally say that Edwards would have affirmed the 1648 Cambridge Platform and the 1658 Savoy Declaration, and that would have been the case for many conservative Reformed ministers in his era, but Edwards had an on-again-off-again relationship with Presbyterianism, including the Erskine Brothers in Scotland (founders of the Associate Presbytery, later the Associate Reformed Presbyterian Church), so there is some legitimate question of whether he was in agreement with Congregational or Presbyterian church government. A lot of his criticism of Congregationalism, if read carefully, can be understood either as a desire to return to local elders in locally autonomous churches (which was the official Congregational confessional position, in which case Edwards would only have been saying, "We need to go back to our confessions"), or in the case of his letter to the Erskines, trying to be polite while politely declining their offer to come to Scotland. Yes, he accepted the presidency of the predecessor institution to Princeton, but that was in an era when Congregationalists were wealthy and numerous and strong, and Presbyterians were weak and scattered with no good ways to train ministers while the Congregationalists already had Harvard and Yale. In the early days, the Log College was more interested in training ministers who affirmed the need for personal conversion than it was interested in church polity. That happened later when it became an exclusively Presbyterian institution, but we can't read the later situation back into the early days when Edwards was named as its president.
To know the answer for certain on what confessions Edwards subscribed to, we would need to look at the official doctrinal standards in effect in the association of ministers to which Edwards' two Congregational churches belonged, first at Northhampton and then at Stockbridge, at the time Edwards pastored there. We would also need to look at the local church covenants in Northhampton and Stockbridge.
Note the term "association of ministers." The very oldest Congregational associations still have formal names like the "XYZ Association of Congregational Churches and Ministers" because by the mid-1800s, the associations of ministers, which typically met monthly, and the associations of churches, which typically met once or twice a year, were beginning to merge.
Those associations often had some sort of local standard for doctrine, and during the First Great Awakening those standards sometimes included specific references to the points under debate, i.e., whether a church supported or opposed an emphasis on personal conversion. There was a widespread belief -- one I do not share -- that the older Congregational confessions, the Savoy Declaration of 1658 and the Cambridge Platform of 1648, did not provide enough specificity to examine candidates on their views of the revival. However, confessional integrity was largely regarded as a matter for local churches to address in their local church covenants, with members usually being required to make some level of doctrinal profession greater than a mere "credible profession of faith" but not full-fledged "confessional membership" as practiced by the Dutch Reformed, a practice the Pilgrims and many other early Puritans knew from their time in exile in the Netherlands. That means without checking local church covenants to find out the doctrinal standards for members and for church officers, it's often hard to know exactly what a specific minister had to formally affirm at a specific time.
The reason this is relevant is that if Edwards had lived a generation or two later, he would almost certainly have been a member of a statewide association of Congregational ministers that used the Westminster Shorter Catechism as its doctrinal standard. That was initially just a happenstance and not intended to be a response to the Unitarians in eastern Massachusetts, but as the Unitarian Schism began, more and more Congregational ministers and Congregationalchurches joined the state association that affirmed the Westminster Shorter Catechism as its doctrinal standard because it gave them the ability to examine applicants for membership and determine if they were or were not in agreement with it.
Note a very important point here. By the late 1700s, a statewide Congregational association was affirming the Westminster Shorter Catechism as a doctrinal test, but NOT the full Westminster Standards. So when you hear that so-and-so Congregational minister in the 1700s or early 1800s "subscribed to Westminster," it does NOT necessarily mean the Westminster Confession, Westminster Larger Catechism, and Westminster Shorter Catechism. In fact, it's much more likely to mean the Congregational pastor involved affirmed the Westminster Shorter Catechism, and used it to teach and in his own church, but did not subscribe to the WCF's teaching on church polity, and may or may not have used the Savoy Declaration or some of its later New England revisions as his confession of faith or his church's confession of faith.
In the early 1700s, being confessionally Reformed was assumed for Congregationalists. The debate was not over the externals of orthodoxy but rather the nature of personal conversion and what level of evidence of saving grace a person needed to present to be admitted into communicant membership, or whether (as Stoddard wrongly taught), that communion was a "converting ordinance."
Those issues are addressed by the Westminster Standards and the Savoy Declaration, and I would say that their teaching is pretty clear, but in the early 1700s both sides CLAIMED to affirm the historic confessions, and the result was that new standards were getting written in local churches and local associations to address the matter in greater detail than the confessions did.
This may sound chaotic, but we have a parallel with the controversies in the 1800s and early 1900s in the Dutch Reformed world in which several different groups of conservatives affirmed the Three Forms of Unity but significantly disagreed on what they did or did not allow.
Given the debate over presumptive regeneration in the Dutch Reformed world, with both sides holding a strict subscription "every article and point of doctrine" view of adherence to the Three Forms of Unity, I think we can see how both sides of the Congregational debate, while claiming to affirm the confessions, believed more specificity was needed. There is language in the Three Forms of Unity to which the Kuyperian supporters of presumptive regeneration could appeal in defense of their views, but the churches in the Nadere Reformatie tradition and much of the Afscheiding tradition were just as adamant that the confessional words didn't mean what the Kuyperians meant by them, which is pretty much the same debate that was raging in the First Great Awakening, though using a different set of confessional standards, on whether the revival was or was not legitimate.
