I think all the extant papal documents were mostly published in the 19th/early 20th centuries by French (and sometimes Italian) editors, so for the most part they're online, at Google Books or the Internet Archive or the BNF's Gallica site:
- Pietro Pressutti, Regesta Honorii Papae III - vol 1, vol 2
- Guido Levi - Registri dei cardinali Ugolino d'Ostia e Ottaviano degli Ubaldini (this is Gregory IX's register from the period before he was pope)
- Lucien Auvray, Registres de Grégoire IX - vol 1, vol 2 (mistakenly labelled as Alexander IV), vol 3, vol 4
- Elie Berger, *Registres d'*Innocent IV - vol 1, vol 2 (or at Hathi Trust), vol 3 (or Hathi Trust), vol 4
- Franco Guerello, Lettere di Innocenzo IV dai cartolari notarili genovesi (a selection of Innovent IV's documents pertaining to Genoa)
- Charles de La Roncière, *Registres d'*Alexandre IV - vol 1, vol 2 (mistakenly labelled as Gregory IX), vol 3; vols. 1-3
- Jean Guiraud, *Registres d'*Urbain IV - vol 1-2, vol 3-4
- Édouard Jordan, Registres de Clément IV - 6 volumes but only vol 6 seems to be online
- Jean Guiraud, Registres de Grégoire X et Jean XXI - [1]
- Jules Gay, Registres de Nicolas III - [2]
- Not sure who edited these, but Registres de Martin IV- vol 1, vol 2
- Maurice Prou, *Registres d'*Honorius IV - [3]
- Ernest Langlois, Registres de Nicolas IV - vol 1, vol 2
- Antoine Thomas, Registres de Boniface VIII - vol 1, vol 2, vol 3
- Charles Grandjean, Registre de Benoit XI - [4]
- These are apparently just edited by the "Benedictines" in general, Registres de Clément V - vols 1-3, vols 4-5, vol 6-7, vols 8-9
For John XXII and Benedict XII there are editions of their "common letters":
- Guillaume Mollat, Lettres communes de Jean XXII - vol 1, vol 2, vol 3, vol 4, vol 5, vol 6, vol 7-8, vol 9-10, vol 11-12
- J.M. Vidal, Lettres communes de Benoît XII - vol 1, vol 2, vol 3, vol 4
Unfortunately this is where my list ends! I think the nature of papal documents changed and probably the later we go, a lot more of them survive. Someone else will hopefully have more experience with Renaissance and early modern popes.
I should of course also mention the ongoing Register of Innocent III project led by Andrea Sommerlechner - Die Register Innocenz' III, which takes up 14 volumes. It should be first on this list chronologically, but it's much more recent (vol. 14 was only published in 2018), and not online.
All of these are in Latin of course, they aren't translated.
There are some online databases that have medieval papal documents in easier to read formats:
Papal Encyclicals - there are a few random medieval documents translated here but they're pretty sparse
ApoScripta has thousands and thousands of documents, but they're not translated. They're organized by pope but it's hard to find anything specific there, unless you already know what you're looking for...
Hopefully that helps. If you're looking for online translations, you won't have much luck there, but most of Latin texts are online.
I can also add, though only a little, to the list suggested below by /u/WelfOnTheShelf:
The majority of the papal letters (at least in the register ) of Pope Gregory VII (r. 1073-85) are available in the book form of either ones:
Selected letters from his registers are also published as The Correspondence of Pope Gregory VII : Selected Letters from the Registrum, trans. Ephraim Emerton, New York: Columbia UP, 1932; rep. 1990.
At least this collection also has a translation of his letter to King Sweyn Estridsen of the Danes (d. 1076), mentioning milites St. Petri and cited by Erdmann.
They also follow the basic numbering of the letters done by Erich Casper in MGH original, so it is relatively easy to check the letter in question, I hope.
There are also the classical registers of the papal documents from 10th to 13th centuries, but there are edited in German (so I suppose that they are missing in the suggested list below). If you are also interested in them, check Jaffe and Potthast in the following bibliography: http://www.hist-hh.uni-bamberg.de/forschung/biblpu.html
Zimmermann, Harald (hrsg.). Papsturkunden, 896-1046. 3 Bde. Wien: Österreichischen Akademie der Wissenschaften, 1984-1989: also includes the text of the letter itself, if I remember correctly.