I'm guessing the answer is no, since the Magic Initiate feat isn't technically the Spellcasting feature.
Spellcasting and Pact Magic are specific class features. Magic Initiate does not count.
When Metamagic Adept says:
Prerequisite: Spellcasting or Pact Magic feature
it is referring to the class features called “Spellcasting” or “Pact Magic” found in the class descriptions of spellcaster classes such as Wizard and Warlock:
As a student of arcane magic, you have a spellbook containing spells that show the first glimmerings of your true power. See Spells Rules for the general rules of spellcasting and the Spells Listing for the wizard spell list.
Your arcane research and the magic bestowed on you by your patron have given you facility with spells. See Spells Rules for the general rules of spellcasting and the Spells Listing for the warlock spell list.
If your list of class features does not have Spellcasting or Pact Magic, you cannot take Metamagic Adept. Magic Initiate does not count as the Spellcasting or Pact Magic feature because it’s not the Spellcasting or Pact Magic feature. It’s the Magic Initiate feature. For it to work, the prerequisite would have to say something like the prerequisite of Elemental Adept:
Prerequisite: The ability to cast at least one spell
If this were the prerequisite of Metamagic Adept, then Magic Initiate would qualify you for Metamagic Adept. Instead, Metamagic Adept names two specific class features as eligible prerequisites, and Magic Initiate is not one of those features.
This answer is completely wrong.
The DMG has an errata p136 that covers what spellcasting means (at least in relation to attunement requirements, however its reasonable to assume this is a universal rule).
'spellcasting' merely requires that you can cast a spell using a class feature or trait. feats are class features, obviously, and even elves who get the one cantrip choice, qualify as spellcasting as its their racial trait.
similarly, variant humans who use their racial feat for magic initiate would qualify.