It is well known that every construction that can be performed with compass and straightedge alone can also be performed using origami, see:
R. Geretschlager. Euclidean Constructions and the Geometry of Origami.
Mathematics Magazine 68 (1995), no. 5, 357–371.
If one checks the paper by Geretschlager above, one sees that the construction for intersecting two circles is quite complex. Is there an easier construction?
The axioms for origami can be seen at:
http://origami.ousaan.com/library/conste.html
Background: I teach a course in geometry for future teachers. With this construction (and other easy constructions), it would be fairly clear that every construction that can be performed with compass and straightedge alone can also be performed using origami.
I only have found two origami constructions for intersecting two circles – one in the above reference and another in a forgotten (!!!) reference. It was a strange book written in the 1950s about various geometric constructions.
Best Answer
Have you found the papers by Roger Alperin? He has some very nice articles, especially in regards to relating various construction systems -- in addition to origami and compass-straightedge constructions, there are a variety of others, like the Vieten constructions, Pythagorean constructions, and even some variants on the origami constructions. Basically, each additional axiom potentially provides an entirely new class of constructions (which may or may not actually enlarge the set of valid constructions). Some of these questions end up being very geometric in nature, and others have some fantastic ties to the structure of algebraic numbers.
In any case, you can find several papers of his via Google. I believe your specific question about intersecting two circles is answered in Section 6 of Alperin's "Mathematical Origami: Another View of Alhazen's Optical Problem."
Hope that helps.