The short answer is that there is data out there, but a lot of it is bad data. If you want good information, you’re probably going to have to actually dig into the raw data yourself so that you really know what is going on.
Easiest example is that a few years ago some media outlet wanted to answer the question of “what is the most heavily armed state” and they did an “analysis” using only the NFRTR (National Firearms Registration and Transfer Record) which only applies to NFA weapons. It makes sense if you are going into the analysis with the “common knowledge” assumption that all guns are registered, but because they didn’t know that their assumption was wrong and didn’t check and fully understand the data, their conclusions were useless.
To your question, the FBI does keep basic stats on NICS checks that are available. In a quick search I didn’t find any detail about type (remember the input they receive is very limited anyway, IIRC they only get “rifle/shotgun”, “handgun”, and “other”) But you can get stats of totals by year, month, and day of month.
Remember that not every transaction requires a NICS check. Of course there are private transactions which aren’t counted, but certain states have CCWs which allow an individual to bypass the check (they still have to fill out the 4473, but the call doesn’t have to go out), and certain states use a Point of Contact where the state runs the background check rather than the FBI. That last group may or may not be included in those overall NICS numbers.
Also note that you can have multiple firearms covered by one 4473 and one NICS check.
NICS checks only give you info on transfers, not ownership. It can sometimes have correlation to purchases, but not always.
Also consider your scope - if you’re interested in answering the question “how many guns are in the state”, a used gun transferring from one resident of the state to another resident of the same state may have a NICS record associated with it, but the gun started in the state and ended in the same state. So the total number of guns in the state remains the same.
Another thing to watch out for is survey data. Generally survey data is a decent (though obviously not exact) way to get a general idea of what is going on, but I would not make that assumption when it comes to surveys regarding guns. There is a very strong (and reasonable) likelihood that people who do own guns would say they do not when answering a survey on a cold call.
There’s also the issue of participant selection. If you’re doing a national survey, the laws of different states are a big factor. For example if you base your survey population on the national population distribution, you would likely have more representation from places like NY and CA which have stricter gun laws and probably lower chances of randomly selecting a gun owner (or someone who will admit to a stranger that they own a gun).
Conversely if you have the opposite bias in sample selection and choose predominantly individuals in rural areas, or you do something like survey everybody on the NRA mailing list, you’re likely to have the opposite problem of your data overestimating the actual.
Then there’s how the questions are worded, and surveys should circle back to topics and use slightly different wording on subsequent questions. This lets the analyst determine if framing (the question structure, but also the questions immediately preceding the actual question being reviewed) impacts the answer.
Proper survey design and sample population selection is a pretty deep subject (think multiple undergrad and graduate level classes devoted to just that topic). Plus once you learn all of that stuff it impacts how you answer surveys as a participant. Well constructed surveys actually control for people who understand this stuff because the impact is statistically significant and can sway the results.
If you’re looking at studies, make sure they are talking about confidence rates, significance in their results, and sources of error. It doesn’t mean the information will necessarily be right, but it at least means they tried to use basic statistics correctly.
Anyway, I know some of this doesn’t directly answer your question, but hopefully it points to why this is a difficult thing to measure and why the information is not easily available.