Please note: this is, as the title suggests, a very incomplete collection. However, all stories, to the best of my knowledge, are both documentably period and freely available online. Consider it a starting place, if you would.
Note also: where a story takes your fancy, I highly recommend seeking out a recent scholarly translation, if translation has been necessary. It can make a significant difference to the text, and the online versions are rarely the best, simply because of copyright laws.
Finally, for those who are interested in retelling these stories, I have attached the handout of a class I’ve run on medieval storytelling.
Prose Merlin (15th c) – a Middle English translation of a 13th century French text
Reynard the Fox (16th c) – this story is originally French, and the earliest versions date back to the 12th century
The Book of Golden Meadows – Masoudi (10th c)
1001 Arabian Nights: I haven’t listed a date, as individual stories were often added to the mix in different versions. If you’re curious, I encourage you to look into the history of this text further yourself
The following websites contained too many sources for me to list them all, but I highly recommend them:
The Middle English Texts Series – http://d.lib.rochester.edu/teams/text-online
The Corpus of Middle English Prose and Verse – http://quod.lib.umich.edu/c/cme/
Icelandic Saga Database – http://sagadb.org/index_az
Luminarium: An Anthology of English Literature – http://www.luminarium.org/
The Celtic Literature Collective – http://www.ancienttexts.org/library/celtic/ctexts/index.html
In Parentheses (a collection of texts and related essays from many medieval cultures): http://www.yorku.ca/inpar/