Is it okay to inquire about the status of a paper when the online submission system shows no update three months after submission?
Yes, you should send a quick email asking for an update. In my experience, every electronic manuscript submission/tracking system seem to be different. Even different journals from the same publisher who use a unified log in system, seem to have tweaks to what you see. I think that after one month at any one stage, with the exception of "under review", it is perfectly reasonable, and probably desirable, to ask for a status update. The time a manuscript spend "under review" is often very field, and even journal, specific and can range from days to 12+ months.

When sending an email, be polite and explain that you just want to make sure that you do not need to do anything at the current stage. In all likelihood the status update will be no more informative than "we are working on it" (i.e., they haven't lost it), but that is informative.