You wouldn't say either "the cause that" or "the cause why". You would say "the reason why i'm writing to you is...". Plenty of people would also say "the reason that i'm writing to you is.."

The expressions "reason why" and "reason that" are common idioms in modern English. "Cause" and 'reason" can by synonyms when referring to a person or agent that brings about or occasions something, or when referring to motivations, or considerations moving someone to action. Examples: She is the cause of all my problems. = She is the reason for all my problems. I have cause to question his integrity. = I have reason to question… There is no cause to be afraid of witches on Halloween night. =There is no reason to… The expressions "cause why" and "cause that" are not commonly used in modern English. There is no grammatical reason why they could not be used. It is probably because the word “cause” when it means ‘reason’ is reserved for more formal use. The only instance of the use of “cause why” that comes to mind is one known to all English speakers…the famous line in the traditional English marriage ceremony. -Therefore if any man can show any just CAUSE WHY they may not lawfully be joined together, let him speak now or forever hold his peace. .