Spells cast using magic items are treated as if you had cast them
DMG 141 states,
Some magic items allow the user to cast a spell from the item. The spell is cast at the lowest possible spell level, doesn't expend any of the user's spell slots, and requires no components, unless the item's description says otherwise. The spell uses its normal casting time, range, and duration, and the user of the item must concentrate if the spell requires concentration.
The relevant passage in the Find Familiar spell states,
Finally, when you Cast a Spell with a range of touch, your familiar can deliver the spell as if it had cast the spell.
Note that both passages here use the terminology "cast a spell" in the first line. In fact, nearly all such items are careful to use this wording. The particular staff you're referencing is homebrew, but it replicates that wording exactly.
The key idea here is that the user is casting the spell from the item. For all intents and purposes, the user is casting the spell normally, as if they had known and prepared the spell themselves. Therefore, because the range of Contagion is touch, your familiar should be able to deliver the spell.