First off, the terms normalize and standardize are both used variably and even unpredictably across different branches of statistical science, and beyond, so bitter experience teaches me that you cannot be confident about what is meant unless the equation, or equivalently the computer code, being used is visible or documented.
It sounds as if you want to scale each company so that some measure becomes (value for company)/(value for "big" company in its state). You can do that, but inevitably you set aside thereby the absolute values concerned. Comparisons, particularly between companies in different states, are therefore made more complicated as much as they are made easier. For example, let's say that $A$ is an arbitrary company and $B$ is the big company that is a reference, so that your measure is $A/B$. Then it is easy to see that (e.g.) in one state $A/B$ could be $2/10= 0.2$ and in another state $A/B$ could be $4/40 = 0.1$. Hence, without paradox, $A$ is absolutely bigger in the second state but relatively bigger in the first state. Whether this is what you want is a matter of your substantive goals, which are not evident and in any case likely to beyond the scope of this forum.
It is very hard to say much about the consequences for factor analysis. Your scaling is just a linear scaling, but a different linear scaling for each state, so all depends on the detail. It could make matters worse or better. There is certainly no sense in which linear scaling is guaranteed to make data behave better. (The literature is muddy here, if only because "normalize" is often used for some transformation designed to bring a distribution closer to the normal, which is often (rightly or not) thought a good idea for methods like factor analysis.)
In general, I often see people in this forum reaching for something like this. My instinct is that often it's a lot simpler in the long run to keep with the original measurements, which are, or should be, on scales you should understand substantively (as a currency amount, a production total, number of employees, or whatever it is). Scaling like this, then analysing on that scale, and then somehow trying to interpret results correcting for what you have done can be a very roundabout style of analysis. It can be true that measures on different scales can be difficult to compare, but there are many methods that offer a solution there.
