Ask your DM
But an Alchemist's Kit does make sense here.
First and foremost, the crafting of magic items is entirely the DM's purview. The rules for how it works are few and far between apart from time taken and cost required.
If your DM allows magic item crafting, the DMG has this to say...
To start, a character must have a formula that describes the construction of the item. The character must also be a spellcaster with spell slots and must be able to cast any spells that the item can produce. Moreover, the character must meet a level minimum determined by the item’s rarity, as shown in the Crafting Magic Items table.
You can decide that certain items also require special materials or locations to be created. For example, a character might need alchemist’s supplies to brew a particular potion, or the formula for a flame tongue might require that the weapon be forged with lava.
Where 'You' is the DM (Chapter 9)
This is further expanded on in the optional rules for crafting a magic item found in Xanathar's Guide to Everything:
To complete a magic item, a character also needs whatever tool proficiency is appropriate, as for crafting a nonmagical object, or proficiency in the Arcana skill.
XGTE Chapter 2
Thus...the DM is encouraged to require a tool proficiency appropriate for creating a similar non-magical item (you can substitute Arcana). So, for the Restorative (Keightom's) Ointment, you are making an ointment. A DM could reasonably assume that to produce a mundane ointment, you would either need to use an Herbalism Kit or an Alchemist's Kit.
So, in your case, with your DM's approval, an Alchemist's Kit makes sense as a tool you may need to create this magic item. But it is entirely up to your DM if you're even allowed to try in the first place, and if you have to go collect some bizarre exotic ingredients to use as a base for it.