Controlling multiple stepper motors using a single driver
As I commented on the question, you're probably better off with conventional relays given that:


their cost scales better as you go for larger/more powerful motors
they are less complicated (since you stated that you've never
designed a circuit before)


Having said that, if you wish to pursue the solid state approach then you're probably aiming at (MOS)FETs for their low resistance (Rds(on)) and low activation current (infinite gain at DC).

There's just one problem: power MOSFETs have an inherent diode in their construction (body diode) that prevents them from switching current both ways as your H-bridge driver requires. To solve that problem, you must use two of them back-to-back. By doing that, you've made yourself a simple Solid State Relay (SSR).

Here's how Texas Instruments explains it on their SSR design document:


http://www.ti.com/lit/ug/tiduc87a/tiduc87a.pdf

Looks like a good starting point for your pursuit.