Is "vast majority" something to avoid?

I think "vast majority" is perfectly acceptable. Go for it.

I suppose I am being quite idiosyncratic, but when I use an adjective in front of "majority", I use these rankings, especially when thinking about elections:

• Vast majority - means almost all or something like 90% or more, but less than unanimous.
• Overwhelming majority - means well beyond any hope of finding enough who are swayable to take the opposite case or something like 75% or more
• Large majority - means an unquestionable number such that there's no point in demanding a recount or something like 60%
• Small majority - means a comfortable margin, but not enough to take for granted or something like 53%
• Bare majority - means you just barely cracked 50.1% and if this were an election and your opponent demanded a recount, you are probably toast, or 50.1% or more

It may not convey >90% to all readers — you have to be quite careful with this sort of usage. However, if the document uses vast majority in a sense that makes it a subset of most, then I think you can get away with it. It’s certainly a useful phrase, but there's always almost all or all but a few.

If you have numbers to support the differentiation between majority and vast majority, I prefer to just see the numbers and usually interpret vast majority as hyperbole.

Also, I would note that majority doesn’t always mean greater than 50 percent, so there is already a lot of inherent ambiguity.

In a technical document I might use it but only if I had the actual percentage. For example

The result was positive in the vast majority (94.3%) of cases.

I would, however, try to avoid it and would probably go with something like

The result was positive in a clear majority (94.3%) of cases.