If nobody is able to delegate tasks to you currently, see if you can work on boosting your skills or getting accustomed to the ecosystem that you work in. Since you're concerned about the company's perception of your performance, you may want to do this with overt permission. For instance, you might send a (brief) email to a decision-maker which outlines the situation and explains what you plan to do. 2-4 sentences might work for this.
If you take this approach, you might not be "productive" in the sense of producing anything that the company will actually end up using, but the goal would be to increase your future value without needing to burden an already-busy coworker.
For instance, if this is a software developer position (my profession, hence the choice of example), you might glance at the bug tracker to make sure there's something on there. Then, send a brief email saying that if there's nothing in particular requiring your attention, you'll try reading through the documentation and familiarizing yourself with the system, and maybe try to fix some bugs. Then, learn! Read up on things, observe how the system works, change things on a local copy of the code, and so on.
For other professions, try to find a similar sort of task (if possible), and do that. e.g., if you deal with spreadsheets often, learning how to use various features of the spreadsheet software in use by your company might be a valuable way to spend idle time.
This does assume a degree of self-learning, however; if you're still in a stage where you need a mentor in order to learn productively, then this approach is likely not the best.