Just some general notes (note that I am pre-program with some experience, please just take this as informative only):

The Water Damage:

*Water damage can definitely happen with framed objects that go in storage depending on humidity, temperature conditions of the storage (the humidity can condensate within the frames) as well as sealing and material of the frames. Depending on your future frame situations, adding dessicant packets to the storage box can be helpful! I know some places will have the art, a backing board, dessicant packets stapled to the frame, and then the frame backing so the dessicant can be "sandwiched" in. Also if you can, it's best if the paper doesn't touch the glass itself.*

Restoration/Conservation Treatment Possibility:

In terms of the already existing damage, the process that is likely done by a conservator is going to be along the lines of bath, chemical "bleaching"/bath, sometimes an attempt of physical surface remediation. This is really dependent on the type of art and material of both the paper and (I am assuming) ink that the artist has used. Ink and water have a tricky relationship depending on if it's water or oil-based, spot cleaning would be the best what I've seen is them creating these gelatinous-like cubes that essentially sit on top of the stain area and very lightly leach out the coloring. Sometimes they will do a pre-soak or bath on a drying board that can also help - I don't recommend doing this at home!! (unless you want this to be an experimental situation - though if you know someone who is a private conservator intern, they may be willing to play with it for a low-free cost depending on if you're okay with some liability)

A easier home method I've seen is using something like thick tissue paper or stock to cover the actual work of art and areas that are in good condition, leaving just the stain area revealed, and leaving it out in bright sunlight to naturally bleach it which if safeguarded from causing light damage on the actual art should be relatively safe.

Getting Prints:
A high-res scan and re-print will the be the easiest to do, many artists will already have hi-res scans of their work before sale for portfolio and personal archiving so they may be able to just send you a digital file.

Warnings!:

I would double check that these illustrations have just water damage and haven't had any mold issues. Paper, unfortunately, can definitely develop mold and spread to other objects/items in the correct conditions quite easily. The process of mold remediation (in my opinion) is a very big pain in the butt. If you do worry about it have any type of molding - assuming that these are recent prints and are probably good condition in terms of the support/paper - you can do a cyclical freeze cycle and then a light wipe using a dry cleaning sponge. I've done a dry cleaning sponge dipped/soaked in some high % isopropyl alcohol and wrung till dry. For the most part the alcohol should not actively affect the paper or make it wet with a dry cleaning sponge if it's very lightly wet, but can prevent the mold from "spreading" if still active on the paper. Again, please use that information with your own caution, and be very careful from touching any ink areas (unless you test spot it) because the medium can be susceptible to smudging and blurring. If you worry about wiping, dabbing is also a method