Just looked into monster servos and linear actuators on pololu.com, and they all cost more than $75. Are there any cheaper options?
First off - SLA (sealed lead acid) batteries are -not- car batteries; while there do exist "sealed" car batteries (that is, ones which you do not need to top off with distilled water every now and then), they aren't the same thing. SLA batteries are also known as "gel-cel" batteries; the electrolyte in them is in the form of a gel, and the batteries can be used in nearly any position. Car batteries, on the other hand (even sealed ones) are designed to only be used in one upright position, and must be secured properly to prevent tipping. The electrolyte inside (whether sealed or not), is a liquid, and can spill.
Regarding servos/linear actuators: If you want this project to be successful, you are likely going to have to spend some money. You could try to cheap out and use something like a window actuator motor, or a wiper motor from an automobile (or even a seat adjustment motor), but such motors aren't designed for the application you are wanting to use them for. As such, you may get lucky and have one work for a while, only to have it fail later (perhaps at a critical point).
Furthermore, understand that the actuators are only one part of the problem; the motor controllers (for the cart motor and the steering actuator) are likely not going to be very inexpensive either; just keep that in mind.
Finally - you need to be aware of, and design in, failsafes - both manual and automatic (and maybe even remote) into your system; for instance, you need to be able to kill the power if the linkage to the steering connection fails, or if the feedback potentiometer goes out of spec (or is disconnected in some manner, or a wire breaks, etc). You need to think about this part of your system *first* - before you build any of the hardware, and before you program any of the software. This isn't something you want to have as an "afterthought" or attempt to "bandaid" in at the last moment. Security and safety design is critical, and should be done at the earliest possible level of a project, to help ensure that it is implemented properly, and without faults.