The answer to your question will depend upon the context in which you've encountered it. Basically, a 'budget share' is exactly what it sounds like. It's the share of an overall budget which an individual or organisation receives from the body which 'holds the purse strings'.
Thankyou Chris! It was in reference to education, i have to write about funding of primary schools for my Health and Social Care coursework, and the link you added is extremely helpful!! =]