I went to the cinema and I'd obviously eaten my pizza too quickly and I was sick while standing in the queue to get popcorn. Lucky most of it went in my hand and the rest on my clothes. I went outside after that and sat on a bench. Threw up 3 times and then left.
Most people don't care if you're sick in public, its a natural thing that happens to everyone. I was once on a train and a woman in front of me was very drunk, fast asleep, with her boyfriend. She woke up and was sick all over herself and the seat. Her boyfriend tried to clear it up as best he could with a plastic bag, it stank but she obviously couldn't help it so I just went and stood somewhere else.
If you can find a toilet, then that's really your best option. Otherwise a bin, in a bush, in a plastic bag. If you really are that worried maybe carry a plastic bag/sick bag with you? and I would use public transport rather than anything else, people are sick on trains/buses all the time.