Does each character gain an additional Spirit Point at fixed levels, or is there a way to help that along? For example, does having a high will help with SP in a way that I could boost a character's stat with the relevant seed and have more SPs faster?
You're correct with your first thought. The amount of spirit points generated by each character each turn only increases when the character reaches a fixed level. It's a unique stat on each character that isn't affected by will or any other character stat.
It can't be increased by any external method and it increases at an exceptionally slow rate. You only get 1 extra point about every 15 - 20 levels in the early game. By level 50 your party will be generating 13 or 14 points each turn (It's been a while since I've played so I can't remember exactly).
The increase also doesn't come at the same levels for each character, Vyse will get additional spirit points a couple of levels earlier than the rest of your party but it'll still be a long time between getting each extra point.