All avenues have to be explored and I would suggest considering untapped opportunities which may exist closer to home within your own community. Don’t be dismissive of local, well-established charities. Talk to ones in your own community and find out what their aims and objectives are under the terms of their constitution - it may well be that they have a small budget set aside which could be available to local schools.
What kind of programmes might charities fund? In my experience, it varies from charity to charity; research locally and look at their websites. Perhaps, as a starting point, try talking to your local councillors who may be aware of criteria for such applications; they may even get involved themselves. I know my colleagues at our parish council also have roles in charities. It's the old case of ’if you don’t ask, you don’t get’, so fire off a letter of approach - you might be pleasantly surprised.
While I was Trustee of a local charity, I encouraged them to consider projects for children within the village as up until that point they had only worked predominantly helping the elderly. A budget was soon approved for the buying of, and, the installation of a friendship bench at the Junior school as part of their ‘anti-bullying’ campaign. It was donated in memory of one of the founder members of the charity who it transpired was also a teacher at the school. The charity now sponsor a small annual award for ‘the most caring child’ where the recipient receives gift vouchers and we also replaced the children’s cricket team equipment that had been destroyed in a local fire.
So, apart from bringing a community together, it is definitely worthwhile asking local charities for help. You never know, they may be able to assist you in purchasing that much needed playground and gym equipment after all.