Stone Casting Likely Allowed, but there is case for No.
The Reserve Ioun Stone only states that:
While this stone orbits your head, you can cast any spell stored in it.
There is no difference in casting language between the stone or a spell naturally, just that you can cast any spell stored in it.
However, there is a case against this with the requirement of expending a spell slot (see below.)
If allowed, this is an excellent way to expand the list of possible Contingent spells outside of the traditional Wizard spell list.
Next is to look at the requirements of Contingency
Choose a spell of 5th level or lower that you can cast, that has a casting time of 1 action, and that can target you. You cast that spell—called the contingent spell—as part of casting contingency, expending spell slots for both, but the contingent spell doesn’t come into effect.
What happens here is that you have cast both Contingency as well as the spell from the Ioun Stone (assuming it passes the criteria of 1 action casting time and targets only yourself).
Do note that this frees up the space in the Ioun Stone as the spell has been cast.
The spell cast from the stone is no longer stored in it, freeing up space.
The contingent spell is fully cast and just waiting to come into effect. There is nothing more needed for the spell to take effect other than the trigger provided.
To answer your specific questions:
Does having the spell stored in such a way count as a spell "you can cast"?
Yes, the language is identical between casting from an ioun stone or casting naturally.
Does the fact that the spell is stored in an item mean that it can't be cast as part of another spell as Contingency requires?
There is nothing to suggest this given the casting language is identical.
Is there any other reason why this would not work?
The main remaining concern is the requirement from Contingency of
expending spell slots for both [Contingency and the spell to be triggered].
If it is required for a spell slot to be used, then using the Reserved Ioun Stone will not qualify as there is no spell slot expended.
I think this is a bit of a stretch, but a strict RAW reading may preclude the use of the Stone with Contingency because it failed to expend the spell slot.
This concern is similar to Ring of Spell storing requiring a spell slot to be expended when casting a spell into it. Jeremy Crawford confirms that here
It requires a spell slot. A wand doesn't expend a spell slot.
Comparison to other Items:
Ring of Spell Storing
The Ring of Spell Storing also uses similar language and should be able to be used in this way as well.
While wearing this ring, you can cast any spell stored in it.
Hat of Disguise
This is an example of something that would not work with contingency. The language here requires an action to cast Hat of Disguise.
While wearing this hat, you can use an action to cast the disguise self spell from it at will.