Your question isn’t much of a WordPress related question, more like a MySQL question.
But I have sort of a solution in my head:
You could go with a custom query and a replace on the apostrophe:
SELECT * FROM `table`
WHERE REPLACE( `column`, "'", "") LIKE REPLACE( 'string', "'", "" )
For a combined search:
SELECT *, REPLACE( `column`, "'", "" ) AS `custom`
WHERE `column` LIKE 'string'
OR `custom` LIKE 'string'
It isn’t worlds most beautiful solution, but you could give it a shot.