The Whitecaps tried to build a privately funded stadium in the mid 2000s in a few different locations before they joined MLS. They were opposed by Gastown businesses and the Port of Authority and gave up. Here's some actual sources from the Whitecaps from back in 2008 as well as an administrative report from the city from 2007:
https://www.whitecapsfc.com/news/whi...ouver-province https://www.whitecapsfc.com/news/van...-remains-dream https://council.vancouver.ca/2007020...ments/csb6.pdf
I seem to recall that BC Place was pitched as a temporary home when they first joined MLS but they're in season 16 now and I guess just now are they trying again, perhaps too late.
Also there's a little stadium just South of the city that is privately funded. Only club in the country to own their stadium and have paid for it in full if I'm not mistaken.