Tomatoes are sort of ok. If you are being super strict about it, then you would want to avoid them. But they are not really that bad. People use the home blood glucose testing, before eating, then 2 hours after eating, to see if a certain food spikes their blood glucose. If someone finds that tomatoes do that, they might want to avoid them. If someone else finds they dont spike it too much, then they can eat them.
Celery is ok I think. Sweetcorn is not low carb, same with potatoes, yams, etc. generally if a vegetable grows above ground, such as brocolli, cauliflower, spinach, it is ok. I'm probably not the best person to advise on this as my own testing and low carbing experience is not that great. There are threads where this stuff is covered in more detail.