I've heard both "I'm finna go to the store" and "He finna go to the store." Do we prefer with "is, am, are", or without? Is it a regional / dialectic difference, or are they interchangeable?
In African American Vernacular English, the verbs is and are usually (not always) get dropped, but the verb am is almost always kept. See for example Wikipedia. Thus, you may hear:
He finna go to the store,
I'm finna go to the store.
This is colloquial.
'Finna' means 'fixing to', its usage is similar to 'going to' or 'gonna', perhaps closer to 'getting ready to'.
If you're trying to use this slang in an otherwise properly constructed sentence then you would say, "I'm finna (fixing to) go to the store", but usage of this slang might go hand in hand with eschewing conventional sentence structure in which case do whatever the people you're trying to fit in with are doing.