Ah yes. The client who thinks that they should be able to edit graphic work after its been produced. There is no technical or tool solution to this problem
If the graphic work is in a source file, to edit it the user will need the same program that created the work in the first place or a similar program that can open, understand, edit, and save the same file format. There is no way around that requirement.
So unless your client is willing to invest time and money into Illustrator (or a similar program which can open Illustrator files), they are better off working with you to create new derivatives based off your original artwork.
So how to handle your particular situation?
I'd see if the client is open to paying a retainer (x dollars a month) so you can generate the changes they want as needed. This way you help control the quality of your design work and can provide the necessary graphics and they can get the graphics they need without having to invest in costly design programs and the education needed to use them to create the graphics themselves.
If they still insist on being able to "do it themselves", I'd allow them the right to purchase the source files. But at this point, our business relationship is over - those graphics are not mine and I probably don't want my name associated with them as the ex-client butchers your designs making changes. Also, at this point, it's the ex-client's responsibility to figure out how to edit and use the design work. They could pay for my time to advise them (but I'll likely be busy working with better clients who appreciate my skill and talents), but otherwise, they are on their own.
So two approaches as I see it.
See if you can convince them that its worth their time and money to pay you a small regular fee to make updates and edits. It really is worth it in most cases - where are they going to find time to become graphic designers and master the tools needed to make their updates? Or the client pays you for the rights to the source files and then its all up to them to figure it out.