After digging around for a while I discovered a sample showing exactly how to do this here.
I am leaving that code here for future reference as it was very useful:
def send_complex_message():
return requests.post("https://api.mailgun.net/v2/DOMAIN/messages",
auth=("api", "key-SECRET"),
files={
"attachment[0]": ("FileName1.ext", open(FILE_PATH_1, 'rb')),
"attachment[1]": ("FileName2.ext", open(FILE_PATH_2, 'rb'))
},
data={"from": "FROM_EMAIL",
"to": [TO_EMAIL],
"subject": SUBJECT,
"html": HTML_CONTENT
})