My wife and I are British citizens who are vacationing in San Jose later this year. Am I correct to assume that as non-US citizens, therefore without a state ID that we would not be able use Reed's Indoor range?

I believe you can utilize their gun range as long as you and you spouse can provide proper identification, in your case, your passports should suffice.

It would not be against the law for you to rent them, as you are not the owners of the firearms. That said, Reed's individual policies might exclude you from eligibility. If you have a photo-id issued by your government (maybe even a passport) that states your age, you'll probably be fine. The only way to know for sure would be to call ahead and ask.