Will my own items' active make my champion cancel my current spell?

I can only think of two item that have a cast time (Hydra/Tiamat) but for those that don't have one you can safely use them while channeling.

As long as an Item or Action doesn't force any movement, casting, crowd control or channeling you can use it while channeling without interrupting the spell.

As Lolwiki states:

Most activatable items and summoner spells can be used while channeling without interrupting, which is particularly useful information for jungle Fiddlesticks: as consuming Health Potions will not interrupt Drain.

Things that do interrupt channels are:

• Flash (Forces movement)
• Teleport (Forces channel + Movement)
• Hydra/Tiamat (Has its own cast time)
• Zhonyas (Forces self stun)

Also a thing why BotrK is pretty popular amongst ADCs like Vayne is the fact that its active doesn't interrupt your Autoattack sequence since it doesn't have a cast time.

You can also use Wards and Trinkets while channeling or being stealthed without breaking these two things.