Access without a car?

Local hotels and the park's visitor centre (which is located on the shores of the Kaeng Krachan reservoir) can arrange drivers to bring you to the Ban Krang camp and when you want to go back you can ask the rangers to call a driver to pick you up. I do not know the going rates, but cheaper than a private tour.
I have never heard of group tour options in the area. 
The cheapest guided options can be found at BaanMaka.com. They offer full day 'guided' options with local guides/ drivers that do not speak English, but know the area and can show you around. In that case you are likely to see more than by visiting on your own. Not sure, I think it excludes entrance fees, and meals., but better to ask them about it.

I am not sure if the rangers will allow you to walk in from the main gate. From what I have heard they won't allow you to do so. But especially if you visit in the weekend, you might be able to hitch hike your way in with other visitors, if you find a way to get to the park entrance (which might be complicated). 

Good luck
  

Yes generally it's the elephants they use as an excuse.
Every park has its own rules and depending on various factors they decide what's allowed or not.
I agree that it is not really a logical reason, elephants are not a common sight in the main area of Kaeng Krachan (road to Ban Krang etc). Definitely not as common as in Khao Yai. But there is a chance. And there was a case of someone being trampled to death by elephants some years back so that does not help the rules for being changed.
In my opinion a more important reason to reject motorbikes would be, because a lot of them are noisy and, generally people on motorbikes do not drive slow. But I know it is unfair to those without modified exhausts and with proper respect for nature. But I am happy they simply reject all bikes to avoid Kaeng Krachan turning into the next KHao Yai where you an be 5kms deep in the jungle and still hear the sound of motorbikes.
And even though the rules say noisy bikes are not allowed in Khao Yai either, it is only enforced when a biker arrives on its own. Seems like bigger teams are always allowed in. Maybe the officers at the gate feel a bit intimidated and do not dare to reject them, haha.
Of course same could be said of cars, but the percentage of noisy cars versus regular ones is quite small.

Anyway, I think the fees for the drop off service to Ban Krang that you can arrange at the visitor centre is reasonably priced and definitely worth it. It helps the locals around the park to make some money, and I believe that is the best way to keep them 'out of the forest'. With that I mean, keeping them from turning to poaching.
It's already hard enough for them now the road to Panoen Thung has been closed. I am sure this has put a lot of locals into money problems, not able to pay the monthly loans for their vehicles. As much as I feel annoyed when the morning convoys raced by, I still feel sympathy for them in this situation.