I know the Alchemical Exalted need to take the Perfected Lotus Drive charm, which permanently takes up one slot, in order to be able to learn martial arts, but do the individual Arts charms themselves take up Slots? Logic dictates they wouldn't- the PLD just stores them in memory, but I can't seem to find this specific phrasing in the book itself.
Short answer: No.
To quote the charm rules text of the Perfected Lotus Matrix; "This Charm fits into a circular port at the base of the Alchemical’s skull, with 10 memory crystals housed in a wheel at its edges, each painstakingly cut to contain one thousand facets. The device enables the Champion to learn, invent and use martial arts Charms by physically housing an infinite capacity library of martial techniques and transcendent epiphanies. Like a Weaving Engine, this Charm may not be placed in an Array and can’t be removed."
The Perfected Lotus Matrix serves as a storage space for all the data and essence patterns needed for any martial arts charm learned. No additional charm slots are needed, nor do any martial arts charms come with an attunement cost for the Alchemical Exalted.
As a note, I'd like to warn either you or your ST that since Alchemicals can purchase their regular spare charms at 1 BP each, it is rather critical to disallow them using any of their eight starting charms as Martial Arts charms, and insist they pay the full 6 Bonus Points per Martial Arts charm they start with.