Looking for nice places to get married in Vancouver, British Columbia,
CANADA. It will be a small private ceremony, at most 10 persons
involved including the bride and groom.

Ideally, a patio or terrace overlooking the sea and/or mountains, or a
quaint little chapel overlooking the sea and/or mountains (I am not
particular about the denomination of the church).

Other suggestions welcome.

I realize this does not fully meet your requirements. But, my wife and
I once took one of those luxury ocean cruises starting in Vancouver. 
We took an optional tour of Vancouver our first day. We were taken to
a beautiful park, I think it was called something like Queen Victoria
Park. Whatever the name, locals will be able to tell you.

This has to be the most beautiful park I have ever seen.

However, of interest to this topic is the fact that many, many
weddings are held there, right out in the open. As we waited for our
bus to start again, a woman in street clothes, who looked to be of
Asian ancestry (not sure that is important though it might be if this
is an ethnic tradition) jumped out of a big limousine, with an arm
full of wedding dress and veil, etc, and ran into one of the restrooms
to put them on. and, it looked like "full dress" except in a park.

We were told that a very large number of weddings are held in that
park right in the open.

The reason I bothered to mention this is because the park is so
absolutely amazingly beautiful most folks need no more scenery.

And, even if the idea stinks, this comment is totally free. :)

Forgot to mention, it seems to rain a lot in this city.  And, I simply
don't remember if they have pavilions in case of rain.