Would the mixture of vinegar and baking soda make a cleaning agent?
Well, cleaning is a very broad subject. This is a question which could make a great Science Fair Project. Because vinegar is just a dilute solution of acetic acid (ignoring trace compounds present), and because both baking soda and sodium acetate are pretty soluble, it wouldn't probably be practical to try to make a saturated acidic solution of BS. That is, any precipitates will be made from alkaline solution, under normal circumstances. 

There's two questions which would need to be answered: 


Does fresh V+BS slurry act the same way as old V+BS slurry? 
Are either better than water + BS? 


Well, hmm. To be honest, IDK, but I'd guess "maybe" for both (which is why it would be interesting to see what happens). Let me explain: 

Acetic acid dissolves soap scum (mostly K, &amp; Ca salts of fatty acids) and dissolves water hardness (mostly Ca &amp; Mg carbonates) So starting off getting rid of some or all of that junk could definitely be a good thing (and acetic acid is only going to be around for a very short time if BS is present in excess). But we know that acids attack metals, so leaving vinegar on metal isn't a good idea...and guess what? Once the BS fully reacts, the acid will be gone! I can't think of any other plausible reason that V+BS would be better than W+BS. 

One other potential reason would be if the vinegar changed the abrasiveness of the BS, making it more effective (which is another thing you could test for), but I doubt it would. (possible but not quite plausible, if you know what I mean). Anyway, what you've got left is W+BS after the reaction, so what's the worst it can do?

I do know for sure that most cleaners (with the exception of many metal cleaners) are alkaline because higher pH tends to dissolve (and/or lift-off (which means form a colloid of or weaken the adhesion to the surface-to-be-cleaned of)) organics better than acidic solutions. Cleanser, which is usually just mostly silica, certainly is more abrasive than BS, but that also means it will tend to scratch (or fog) more surfaces. I don't know, but I'll bet most cleansers are alkaline for the reason mentioned above. (alkaline solutions are NOT more slippery, but they form soaps out of your skin, and those soaps are more slippery. So, the slipperiness you feel when you touch an alkaline (basic) solution is just your skin dissolving :)