When you say you have applied, have you actually submitted a completed application, or just telephoned or emailed them?
Was going to ask exactly the same thing. You need to make a formal application, using whatever the system is locally (using two examples not in your area, Essex only handles mid-year applications for community / VC schools and all other applications must be made to each school individually, whereas neighbouring Thurrock handles all applications centrally via a CAF- you need to check). Only then will you be formally offered / refused. If refused, this triggers your right to appeal and you can (should) do this simultaneously for any schools you want.
Schools which process their own mid-year admissions must inform the LA of all formal applications and of the school's response. They don't have to tell the LA every time someone just rings up or emails with an enquiry, however. If you have genuinely formally applied for and been refused by multiple schools, then the LA should find you one.
This is assuming that you have moved into the area, or moved a significant distance within the area, from your previous address, though - the LA won't look for the nearest school with a place, or invoke the Fair Access Protocol, if you have just moved, say, 2 miles, or haven't moved, but just want to change schools?
Is your DC currently out of education?