The manufacturer sells said "vulcanizing solution" in quantities of 25g tubes through 1 gallon cans, so you should be able to buy it in larger quantities. Indeed, a quick search of Amazon yield 8 oz cans (or slightly cheaper), and while I couldn't find it on something well known like Amazon there are other places that sell the 1 gallon quantity. However, as many of the reviewers on Amazon point out, an 8 oz can should be a lifetime supply, and won't dry up nearly as quickly as those little tubes.
As far as what it is, that's a little harder to figure out. Checking the MSDS, we find only that it consists of 40-70% trichloroethelyne, an organic solvent. That doesn't really solve the problem, as that's just the solvent used.
To address the question of rubber cement, if we follow your link to Wikipedia, we find that it's generally a non-vulcanizing cement, while the tire patch stuff definitely claims to be vulcanizing.
Moving on, we can think about what the traditional solution to closely held trade secret formulas is. And the tool that was designed to solve that, though only does so imperfectly, is the patent system. Maybe we can find something in a patent filing.
A quick check over on Google Patents yields something promising, Cold self-vulcanizing rubber compositions; however, upon reading the patent, it's a three part mixture, and the process involves things like "the rubber used is a natural rubber, partially depolymerized by mechanical working and peptization with diorthobenzamidophenyl." I don't know about you, but peptization with diorthobenzamidophenyl is not exactly something I feel like trying to cook up with common household substances in my basement.
Reading further on, we find some further promising patents like Tire repair with cured patch or Tire repair by "patch only" method, but they all also involve discussions of the following form:
Suitable vulcanization accelerators include, but are not limited to, thiazoles, sulphenamides, guanidines, thiourea derivatives, amine derivatives, and combinations thereof. Examples of these include, but are not limited to: N-cyclohexyl-2-benzothiazyl sulfenamide (CBS), dibenzylamine (DBA), N-N'-diphenylguanidine (DPG), Dicyclohexylamine (DCHA), 2-mercaptobenzothiozyl disulfide (MBTS).
As well as references to entire books on rubber and the process of vulcanization: 'Rubber Technology, edited by Maurice Morton, Chapter 2, "The Compounding and Vulcanization of Rubber," 3rd edition, 1987'
None of these appears to have any kind of description of a single readily available material or combination of a few that could be mixed together to work; they all appear to be fairly intricate, specialized compounds consisting of a variety of ingredients.
So I think that the 8oz can on Amazon is probably your best bet. I've heard some people claim you can just use regular rubber cement, but given that there are big cans of the "vulcanizing solution" readily available, my instinct would be to go for that.