I was a bit confused by what you were asking at first, but yes you can do that. There are even a few traditional cheeses that do (whose names escape me at the moment). I've heard of it most often with making whey cheeses (similar to ricotta). The typical thing to do is to take some sweet whey (from making cheese), add a bit more milk to it, heat it to 92 C to make the cheese, drain, cool, salt and form it. Then you wash it with more sweet whey (that wasn't heated). Finally, you age it for a few months. I've done it one time before and it made a very nice cheese. Similar to a ricotta salata, but because it ends up much more acidic you don't need nearly as much salt. I would say it's a very challenging cheese to make, though.
Generally speaking, traditional cheeses aren't made with vinegar or citric acid. Instead, you often use whey that was drained from a rennet based cheese. Because that cheese has been fermenting for a few hours before it's drained, the whey is slightly acidic (though still called "sweet whey"). There is still a fair amount of whey protein in the whey which you can get out by raising the temperature of the whey. It's common to add up to 15% milk to the whey. The goal of the added milk is to hit an overall pH of about 6.0-6.1 and to give you just a bit more yield. The extra casein from the milk also gives you a more firm texture in the final cheese in my experience.
The other way you would traditionally make a cheese like this is to simply add your culture to milk, let it go until it's starting to get sour, but before it makes yogurt. Then you heat it up. An example of this is called Ayib (where you wait long enough to sour the milk to get cheese at a temperature of about 55 C). However, you can make really excellent "whole milk ricotta" or something like anthotyro (a Greek cheese). Or you can make some yogurt, drain it and then use the whey to acidify your milk (you need a *lot* :-) ).
Using the lactic acid that is generated from the culture in milk is a massive upgrade in flavor compared to other kinds of acid, though it is tricky to do (and requires a bit of practice).
If you sour the milk enough and thereby coagulate the curds at a low enough temperature, you don't even need to add more culture to get it to acidify down later. For example, normal greek yogurt (or any main thermophilic culture) will survive temperatures up to 50-55 C, so it's quite possible to make cheese that way and just let it ferment out after you have the curds. Again, I've done this a few times. It's not my favorite way to go as I find that for some reason the cheese ends up being drier than I like.
Finally, the elephant in the room: You can take a mesophilic culture, add it to milk (possibly with a drop or two of rennet, but it's optional), let it sit for 12 hours, then drain it and make cheese that way. Lots of fresh goat cheeses are made this way. Add some geotrichum and you have a choice of other bloomy rinds to call it. Add some penicillium candidum and you have something similar to Brie de Melun. Obviously there are more techniques you need to know about, but no other ingredients.
There are a crazy number of cheeses "lactic cheeses" (cheese formed from lactic acid from the culture you add) you can make that you can age. There is some discussion in the lactic section here, though I don't think that many people ever try it so there isn't *that* much info around.
