Is it ok to show up to a conference without registering if I don't eat any meals?
No, it is not acceptable.
You can of course email the organisers and ask if you can participate without paying, but do not be surprised if they say no.
(Please note that conference registration fees cover lots of things besides the lunch. Among others, conference registration fees may cover the rent of the hall in which you are sitting, and expenses related to the speaker who is giving the talk. Conference organisers are definitely not expecting random people to come there without registering, unless they explicitly advertise this possibility.)
People are discussing whether its OK or not. I'd rather respond in context of career impact. The risk is that the people who invited you would think of you as a "schnorrer", and your future invitations may dry up.
Also, many conferences employ professional conference organizers. If there is a chunk of unpaid attendees, you may impact the relationship between the community holding the conference and the paid organizer, making it a little more difficult for that conference to be held in the future.
If you can pay for one day, do it, and be thankful that the conference had a mechanism that allowed you to do so.
Sometimes, though, if the event is internal to YOUR university, organizers make accommodations for local faculty and students to attend at discount or no cost. Good local attendance can enhance the reputation of a department. If this applies in this case, you should talk to the conference organizer.
Tl;dr: You should ask your mentor what the particular policy and culture at the conference you are attending is.
At least in the humanities and social sciences, it depends.
A. There are many conferences where registration is heavily monitored and you cannot audit sessions for free. These tend to be conferences that rely on registration fees to pay for their hotel/facility costs or the running costs of the sponsoring organizations.
B. There are conferences that would prefer it if people registered/paid, but otherwise do not enforce registration to attend sessions.
C. And then there are some conferences that are entirely paid through internal and external grants and not conference registration fees, so they are entirely open to the public (although some sessions and meals may be restricted).
D. Finally, there are also some communities of scholars that think that type A conferences are morally bankrupt, and so actively encourage their students to "borrow" their name tag / registration badges (which leads to all hilarity during the after-panel social interactions): "I always imagined you as ..... um... older... Dr. Goodall.... um.... congrats on your transition?."