I don't think you're going to find a single, centralized database because digitization is a pretty localized activity. By which I mean each library/museum/university/archive/what have you digitizes and hosts documents from its own collection in most cases. Your best bet for a large repository in the US would probably be the Library of Congress digital stuff, but even there you'll have to search through different collections to find what you're looking for. The same is true of the New York Public Library. I'm not sure about other countries, but I would guess that most have some equivalent of the LoC that will have digitized correspondence from within the country.

Also, many universities have LibGuides like this one to make it easier to find things like this. Semi-related to that, searching google for "digitized correspondence" turns up a lot of university held collections that might not be listed elsewhere, but they're generally going to be pretty small as well. Googling "digitized collection" comes up with things from a wider variety (some universities but also non-university libraries and archives), though again, they're mostly smaller collections.

That's about it as far as I'm aware, unfortunately. Maybe someone else will know of different avenues.

Your quest for one big pool of everybody will lead you instead to a lot of ponds. Yes, there are databases of historical letters online, but tagged to specific people and things. Benjamin Franklin's papers, including letters, are here, and if you search you'll quickly find the website for other Founding Fathers . English poet and diplomat Matthew Prior's letters are here. The National Library of England has posted a selection of letters from WWI here.

You could narrow your search terms: start with names of people or events, add "letters", and see if you get some better results. But I think most researchers here would shake their heads at the thought of anyone limiting themselves only to letters available online. For example, There's letters of the Verney family here. But that's only to 1639, and the actual collection is much bigger.

Notice that, also, a number of scholars have spent a lot of time immersed in the Verney letters, figuring them out. That's because letters are usually primary sources, and unless you acquire some knowledge of the background of primary sources, you will be puzzled . Can you immediately decipher " Have Jemmy tell Mrs Williamson there's no sweepage", or " the Lord Randthorpe problem has us all enthralled" , without knowing something already? You may think you're starting some broad research, but you might only be looking at the entrances to a lot of rabbit holes. You will likely find that you need to first know something about the rabbits.

As has already been mentioned, you'd have to look for a specific topic/time period, because there are probably hundreds of thousands of surviving historical letters.

For example there is an online collection of medieval letters written by women: Epistolae

There are also published collections of letters written by medieval popes. I don't think they're in an online repository, but the publications are generally online on the Internet Archive.