This is not a feature in osTicket at this time (but you seem to know that).
It might be easier than I think at the very least you would need to use a library to decipher the file format of each attachment (word, excel, pdf, etc) and then change the search routines to search through the files. That seems like a tall order but I'm not a dev. You would also probably want to cache the data in the search cdata table for faster search later, or add a separate attachments cdata table to cache the data in.
