I would say no; a season should only magnify shards that you already possess, it could even be a percentage based modifier rather than acting as a full shard level. You're not gaining mana from the seasons like you do from a shard, so you shouldn't be able to cast spells of that particular element type without access to that shard.
Either way, things like this are difficult to justify, because it adds a level of complexity that doesn't necessarily add to the enjoyment of the game. It would simply be another set of rules to try and consider, instead of adding an interesting and fun mechanic to the game. Travel could be slower in the winter to mimic the difficulty of warring in the snow, except, that would just be frustrating and not much fun. Spring could be the "Social Season" except that just means you would skip through turns to get the bonus associated with diplomacy in the Spring.
If something like this were implemented though, I think the magic system would be the best suited for it, because it would make sense in terms of the lore, or would at least be close enough to be able to tie it in without much fuss. Plus, it would make sense on a logical level, of course it should be easier for a Fire Elemental to be stronger, or take less effort to cast in the Summer than in the Winter. The only issue with this too though is that the seasons are very short, so the benefit of having a Fire Elemental have a bonus during the summer would be marginal.
