How can I electrically switch between taps on a transformer?
You can use a solid state relay with zero-cross circuit and a multiplexer to enable each SSR of one single tap. If you turn off the drive signal of the conducting SSR in the middle of AC cycle, the SSR will continue to conduct until the end of AC cycle, meanwhile when turn off the signal of 1st SSR you turn on the 2nd SSR. But the 2nd SSR will not turn on until zero-cross - at begining of next AC cycle. Such way there is almost impossible that two SSR will conduct at the same time.