I translated this as 'She is having coffee' - which makes sense in English. Like when you're at a cafe.
'What are you having? I'm having coffee'.
The only thing I can think is that the Polish don't use 'to have' in this circumstance, and rather use 'to order' or 'to drink'.
Can anyone confirm this?
Ona ma kawę means only that she's in possession of coffee and doesn't necessarily mean that she's drinking coffee.
"kawa" is feminine. All feminine nouns that end in -a in Nominative (so almost all of them) get the -ę ending when used in Accusative. The verb "mieć" (to have) needs Accusative.