Why the gap? Can the furniture be modified? Is the plumbing the issue?

Are you talking about the back of the vanity or the sides of the vanity? If you are talking about the sides of the vanity, I also have a gap on both sides of my vanity because I set mine in an alcove. See below:

How does it work if there is a gap along the back of the vanity? Wouldn't all kinds of junk fall down there?

I draw up everything a million times before I actually lift a finger to do something and I have a similar scenario. We have a free-standing vanity that is only a few inches from the wall it's next to and I don't think it looks odd at all... I think 5" is plenty of room for it to actually look free-standing. Here is one option that I drew for my own bathroom (when we were thinking of keeping the original cast iron radiator in place and just painting the radiator cover for it). There is about 5" to the side of the cabinet and 4" to the side of the vanity top.

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As you can see really the biggest problem with having it so close is not the vanity itself but the sconces are a little cramped. If you're not planning to do a medicine cabinet flanked by sconces that is so close to the width of the vanity then it will be fine (if you are doing a light over your medicine cabinet it would definitely be fine).

Here's another option where we replaced the radiator with a myson towel warmer/radiator combo. I think it looks better, but because there are no longer two "cabinets" side by side and because the sconces have more breathing room. The vanity cabinet itself had plenty of room in the first I think.

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