To determine the grammatically correct sentence, we need to analyze the subject-verb agreement and appropriate use of pronouns. Let's evaluate the given options:

1. Option 1: "It is I who am responsible for this fiasco."
   • The sentence correctly uses the pronoun "I" as the subject complement (not "me" or "myself"), aligning with formal grammar.
   • The phrase "who am responsible" follows because "who" refers back to "I." Therefore, the verb following "who" should be the first person singular "am."

   This option is grammatically correct.

2. Option 2: "It is myself who is responsible for this fiasco."
   • The use of "myself" is incorrect because it functions as a reflexive pronoun, not appropriate as a subject complement.
   • Incorrect pronoun usage makes this option grammatically incorrect.
3. Option 3: "It is I who is responsible for this fiasco."
   • While "I" is correctly used as a subject complement, the verb following "who" should match the first person "I." Therefore, "am" should replace "is."

   This option is grammatically incorrect due to incorrect verb form.

4. Option 4: "It is I who are responsible for this fiasco."
   • The use of "are" is incorrect because the verb should correspond to the first person singular "I," which requires "am."

   This option is also grammatically incorrect due to verb disagreement.

Conclusion: The correct sentence is Option 1: "It is I who am responsible for this fiasco." This choice correctly uses both the pronoun "I" and the corresponding verb form "am."