You can append using to_csv by passing a file which is open in append mode:
with open(file_name, 'a') as f:
df.to_csv(f, header=False)
Use header=None, so as not to append the column names.
In fact, pandas has a wrapper to do this in to_csv using the mode argument (see Joe’s answer):
df.to_csv(f, mode="a", header=False)