I was talking to a friend recently and we were joking about that. He was asking me if someone had ever questioned us about our prints being free of animal by-products and we had a good laugh about it... I have to tell him that it actually happened to someone.
About the question, I would tell the vegan client that, as far as I know, the inks are made of mineral or synthetic pigments, glycols and water and most papers we use are made of cotton with addition of calcium carbonate and potato starch as a sizing agent. What else they may use on the inks or the paper or the coating we don´t know. Some art papers (like Arches Aquarelle) in fact use gelatin as a sizing agent, but I know for sure they do not use it on the Inkjet version of the same paper. I was invited to their factory last year and the person in charge of their Fine Art paper´s production was showing me the gelatin sizing process on the art papers as I asked about the inkjet version. She told me that they tested it but it did not add any benefit, only cost to the inkjet papers, so they do not use it.
So... I would say that as far as I know our prints are free of animal products, but it is hard to be 100% sure.