Would there be any way to get a system in place that could track the virtues that a character is interested in as well as which points they have completed so as to be able to find other members with those needed virtues as well?

If you are not familiar with LotRO, every zone has several deeds that when completed provide virtues that you can slot for bonuses to your stats.

Would this be something that could feasibly done with the custom fields?

Well, what I was thinking involved tracking each of the actual deeds associated with each trait (multiple deeds per trait) so as to check off which ones are complete giving an idea to the other members which ones they had in common to work together with.