Need help with my project

I need a robot with the following specs to complete my project:

1. Strongly prefer width of less than 3.5", the smaller the better
2. Able to push 2X4 wood block of about 4" in length. The wood block will be in a tunnel, so there is no concern about side way movement.
3. Simple linear movement only, and only need to push the wood block repetitively about 3.5" each time, but speed and interval must be programmable
4. Prefer ready made to complete my project sooner

I am thinking there must be programmable toy cars/bulldozers that can do the job, but I can't find it so far.

Please help. Thanks.