Is this a new trend? Paper-free weddings?

That may have been a sweet idea, but the message was inappropriate and tacky, particularly if you got a paper invitation. You mail the check or gift or drop in at their house after the wedding. Just don't use your car to do it.

Few guest bring wrapped gifts anymore. But no cards or envelopes!

That is not a trend. And mentioning gifts is a huge faux pas,

Something inside of me would really tempted to give the cash in a wrapped box , wrapped in a wrapped box, wrapped in a wrapped box with a bow,

I don't think they meant no gifts at all. Because they could have just said that (I did on our vow renewal/marriage blessing invites, to make it perfectly clear). However, without wrapping or envelopes, it will be pretty much impossible to tell who gave what. They probably don't want to be bothered with having to send thank-you notes. Can't send a note if you don't know who gave you what.

You can always wrap it in a recyclable box and wrap the box in a recycled brown grocery bag...... Otherwise you can take the check and hope they have a box where people can deposit their 'gifts'. As others have stated it's rude to mention anything about gifts even if they had good intentions.

no, or at least I hope not.

An gift giver gives what they want and wrap how they want.

There are just things couples have no say, what an gift giver gives and how they wrap it, is up to the gift giver.

I would assume they are asking for no gifts.

So they're probably trying to say "no gifts" in a more polite way (but it's still rude).

I think it's rude.