Learn Hardcore Icelandic: Mig vantar mynt.

Because að vanta is an impersonal verb that takes the experiencer in the accusative case (a “quirky subject”). So you say Mig vantar ... (“Me-ACC is lacking ...”), not Ég vantar .... Use mig because it’s the accusative form of “I.”

What case is "mig," and what are the forms of “I” in Icelandic?

mig is accusative. The four case forms are:

Nominative: ég
Accusative: mig
Dative: mér
Genitive: mín

Who is the subject here? Why is the verb in 3rd person singular (vantar)?

There’s no nominative subject; this is an impersonal construction. The finite verb stays 3rd person singular by default. You can also include a dummy subject: Það vantar mynt. Fronting the experiencer gives Mig vantar mynt, but the verb doesn’t agree with it because it isn’t nominative.

Can I say “Mér vantar mynt”?

You’ll hear Mér vantar ... in colloquial speech, but standard grammar teaches the accusative: Mig vantar .... Stick with accusative (e.g., Þig vantar ..., Hana vantar ...) in writing and in careful speech.