The cartoon does not and likely never will develop a canonical answer for where Dexter gets his resources because to do so would counter and confuse the visual rhetoric message of the cartoon. The cartoon medium is an art form which circumvents our normal reality-based expectations and places perceptive concepts in place of tangible elements. As a quick example, look at Dexter's left hand and notice it only has 4 fingers. If this were a live action character that would be terrifying, as would the proportions of his head and body. Our minds easily suspend disbelief in visual rhetoric, simply because the expectation of realistic proportions does not exist. You can further notice that the glass beaker is not transparent, he has no eyelids, and his ear is on the back of his head. What is important here is that if no one ever mentioned these things, the audience would not even notice. They see a normal hand, a normal ear, glasses, a glass beaker, and so on. That is the communicative advantage of the comic medium, the audience has no expectation of realism because our mind "fills in the blanks" for us through semiotic processes.
This suspension of disbelief carries beyond the visual rhetoric and into all elements of the medium, allowing a message to be tightly focused without distraction. Realistic resource limitations, logistics, and physics actually become distractions in comic medium and generally slow down the communication. Consider the cartoon Ren and Stimpy which occasionally featured realistic drawings and photographs within the animations. At those point, the program had to focus on the realistic visual for several seconds so the audience could adjust their focus.
One classic example of a cartoon's dismissal of limited resources is Warner Brothers' "Acme Corporation," which was an endless supply of any resource a character needed to communicate their message.
The comic medium uniquely allows the audience to gloss over realistic concepts like "How much did the balls cost" and "How did they deliver them so fast?"
In the case of Dexter the focused message is the boy-genius who can solve any problem with his intellect. The message would be completely lost if the program veered off into logistics of how and where he got his money. This is specifically because his audience doesn't have a grasp of money and economy as an integral part of life.
While this may not be the adult answer you were looking for, it is the reason he doesn't need imaginary money to buy his imaginary stuff - the intended audience simply does not have an expectation of economic feasibility, and our minds don't have an expectation of perceptual accuracy. Details are destructive in this medium.
Thus, Dexter get's all of his resources from the Cartoon Network equivalent of the Acme Corporation. For the good of the message, it just is.