Yes 'sickness' here is being used to mean 'vomit' (I see 'vomitus' is in the dictionary, but I have never heard anyone say it other than perhaps a doctor).
Technically, 'sickness' does not mean 'vomit', according to the dictionary. It means:
a) A particular illness or malady.
b) The state of being ill.
c) Nausea, queasiness.
None of these definitions include the actual act or product of vomiting.
However, the tendency of English speakers to use words which avoid directly describing an unpleasant act or entity when in formal contexts (euphemisms and circumlocution) is being used here to avoid directly saying the noun 'vomit' or 'sick'.
While these sorts of paper bags are universally called 'sick bags' in normal English, it's exactly the sort of thing which often won't get written on them.
I just did a quick Google, and the most common sentences used on the aeroplane bags are "for motion sickness" and "waste bag", both of which avoid saying directly what everybody knows the bags are for.