How can I contact LEGO set designers?
Considering individual designers would need to ask for permissions from LEGO to do this, it may be more efficient not to contact them directly, but to go through the community team.

If that doesn't work, it's true that some designers do have an active online presence (check the answer to Are there any great LEGO designers that are well-known?) and shouldn't be too hard to get in touch with. There's always a chance they'll have to send you back to the community team or a legal team somewhere, but I suppose it's worth a try.

Anyway, we need them to spend their time on designing sets, so let's hope you don't demand too much of their time.