Finishing your work earlier than expected (while still performing well) is seen as something nice. Your supervisor will be happy with this!
What I would do, is send a quick email to your supervisor (and possibly your manager too, depending on who keeps an eye on whatever you do.) telling him you finished early, solved the problem and will be asking your coworkers for more work.
I finished task X early and solved the issue by doing Y. I will be asking my coworkers for new tasks to do now, as we agreed upon before you left.
Good luck with the business trip (or holiday, or whatever they're doing)
Kind regards, mynameissecret
This way, your supervisor knows what you're up to. If your coworkers fail to give you any new tasks, you can get back to your supervisor on this.
As for asking for new tasks from your coworkers, I would ask in person first. This way you have the best chance on getting a response fast. I like to start the conversation in person and if needed explain what my skills are. It might happen that they don't know of a task just by the top of their head. I then ask them to email me with any tasks they have available for me.
Now it can happen that they forget to send you a task, or maybe you just don't hear back from them. Send them an email at the end of the day asking for a task.
Did you manage to find the time to assign some tasks to me? If not, do you know who else I could ask for a new task?
Thanks for helping me,
Now, on to what you should be doing while waiting for a new task. As I understand, your supervisor did not give you any material to work on other than the first task. I would put time in learning new things. Read up on articles, search for tutorials etc. If you're unsure if you are allowed to do these things, you could include a line mentioning this in the email you initially send to your supervisor. This way you're letting him know you're doing something productive while waiting.