"on the shoestring"
The film was made on a shoestring.
It is possible to make good movies on a shoestring budget.

Do we add budget after shoestring in writing?
as the word itself means low budget

Question

"on the shoestring"
The film was made on a shoestring.
It is possible to make good movies on a shoestring budget.

Do we add budget after shoestring in writing?
 as the word itself means low budget

Answer

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.

Answer

'on a shoestring budget' is actually fine.