My general breakdown for VS Flying Club members:
- Searching for SkyTeam excluding Air France KLM - use KLM.com (ignoring any flights/itineraries which include AF-KLM)
- Search for AF-KLM on Delta.com
- VS and DL on VirginAtlantic.com
- (You didn't ask but...) Star Alliance partners I would search on AirCanada.com (using the filter function to only show SQ, NH, NZ)

When you found the space you're looking for, use the live chat function of VS website or call up flying club.

Reasoning:
Slightly strange logic - but much of AF-KLM and DL's award space on their respective websites are only available to their own members (thus the ultra high award costs which are variable and not fixed beyond the initial saver space).
DL's website is pretty patchy on SkyTeam partner availability. AF-KLM is generally a bit better.

I have managed to successfully book a flight on China Airlines (CI from Taiwan) - but with a few problems which you can read here Seat selection on China Airline award ticket

Final aside: I did also do a YouTube video on VS Flying Club basics - but the rules on FlyerTalk don't allow direct link sharing for personal blog/YT content (I think) - but you can find a video guide by searching 'an introduction to Virgin Atlantic Flying Club' and skipping to 11 mins or so - it's only about a week old. Other videos are available.