Is it possible? The cards I don't want to buy are fetches, bobs, goyfs, and Lilys.
Not really. You could, but it wouldn't be competitive. There is a reason those cards are as expensive as they are. If you want to try it, I recommend {Desecration Demon} and {Vampire Nighthawk}.
Alright, let me rephrase my question: is it possible to make a semi budgeted version of the deck and then acquire the big things mentioned above later?
Well yes but you could make a budget vintage deck and acquire your moxed and lotuses later as well. Will it be competitive or powerful until you get those ~20 crucial cards? No