I want to have a campfire on Southport beach in a few weeks as a small birthday celebration with some friends. I was wondering who I should contact to request permission to do so, as I know it's illegal to start a fire in England without the land owner's approval.
The land between the high water mark and the low water mark is owned by the crown (crown reserves) in UK law. (ref) The land above the high water mark is owned by land owners, this may also be the crown if it's common land etc.
Any land owned by private land owners is subject to the land owners themselves. They could allow or disallow bonfires at their discretion. So if you want to build a bonfire here you should contact the landowner.
The law on bonfires on common land will be covered by the local by-laws of the council in question.
If the land is privately owned, you'll have to ask the owner.
If the land is publicly owned, you'll have to check local by-laws and/or check with local authorities. I'd assume that in such a case the police should be able to either give you an answer directly, or at least point you to whoever can. (Just in case this isn't obvious: don't call the police on their emergency number, but look up the non-emergency number for the local police station online.)