Note 1: 4E’s Invoker was actually designed around a character who had a literal shard of the divine inside them, and powered their spells off it. 4E’s Shaman class was much closer to what you want here; though they only had a single spirit companion, it could channel a variety of effects, and if you wanted to describe each spirit-effect as a different spirit, it would work without any real problem.
The first thing I do when thinking of a new magic system is to think of its usage rate. How do you see their magic being limited?
If they can only call on each spirit once per day, then Jedavis’ suggestion sounds ideal to me; you just have to alter the standard repertoire system a little bit to account for the specific requirements of each spirit, probably with some kind of a bonus to the class involved to compensate for adding more weaknesses to the class.
If they can call on them multiple times, but at increasing difficulty as they call them more and more, I’d look at the truenamer from 3.5’s Tome of Magic for inspiration. (The truenamer had severe math problems, especially when interacting with other 3.5 elements like ‘magic items that give you a bonus to skill checks’, but the concept was great.) Basically, this would be a system wherein you have to make a throw to cast a spell, and the difficulty for the throw increases each time you successfully cast that spell, until the next day when it resets. (The truenamer actually reset each encounter, and you could of course make it reset at whatever timescale you wish. I just like day.)
If they can call on them until their energy runs out, you’re looking at a spell point system.
If they can call on them at a flat difficulty all day long, you’re looking at a flat throw system, probably similar to what I came up with for psionics/occultist; a target value based on level, with a penalty for the difficulty of the effect being attempted.
Once you know the usage requirements, you can figure out the basic mechanics of the spellcasting, and once you have the basic mechanics, you can implement specific elements (in this case, the individual spirits).
Another way to do it would be to make them not a spellcaster at all, but rather, a class that has a variety of spell-like abilities. You’d just build the class with a bunch of Thievery points and trade them all off for spell-likes unlocked at various levels. In order to make the class more varied, you could either go with paths (like witch did; you might end up with, for example, the Winter path who unlocks cold, hunger, and darkness spirit summons, or the Summer path who unlocks fire, strength, and light summons).