There is no specific, definite "shoestring" so never the definite article "the" Always "a shoestring". "budget " is optional. The expression is so commonly known that "on a shoestring" is used with nothing other than budget, so budget is implied, but still allowed. So both your sentences are viable. Also a shoestring of a budget, describing the type of budget.

'on a shoestring budget' is actually fine.